ALBERT

Description: 〈p〉Publication date: 1 October 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 266〈/p〉 〈p〉Author(s): Richard J. Smith〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In recent decades, topology has come to play an increasing role in some geometric aspects of Banach space theory. The class of so-called 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msup〉〈mrow〉〈mi〉w〈/mi〉〈/mrow〉〈mrow〉〈mo〉⁎〈/mo〉〈/mrow〉〈/msup〉〈/math〉〈em〉-locally relatively compact〈/em〉 sets was introduced recently by Fonf, Pallares, Troyanski and the author, and were found to be a useful topological tool in the theory of isomorphic smoothness and polyhedrality in Banach spaces. We develop the topological theory of these sets and present some Banach space applications.〈/p〉〈/div〉

Description: 〈p〉Publication date: 1 October 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 266〈/p〉 〈p〉Author(s): Dekui Peng, Wei He〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We show in this paper that for a non-compact LCA group 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mo stretchy="false"〉(〈/mo〉〈mi〉G〈/mi〉〈mo〉,〈/mo〉〈mi〉τ〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉, 〈em〉τ〈/em〉 has exactly 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si134.svg"〉〈msup〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈mrow〉〈msup〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈mrow〉〈mi〉k〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉G〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/mrow〉〈/msup〉〈/mrow〉〈/msup〉〈/math〉 predecessors in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si102.svg"〉〈msub〉〈mrow〉〈mi mathvariant="script"〉G〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉G〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉. This answers a problem posed in the literature affirmatively. Denote by 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si195.svg"〉〈mi mathvariant="fraktur"〉N〈/mi〉〈/math〉 the class of all LCA groups 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mo stretchy="false"〉(〈/mo〉〈mi〉G〈/mi〉〈mo〉,〈/mo〉〈mi〉τ〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 such that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.svg"〉〈msub〉〈mrow〉〈mi〉P〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉τ〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 is precompact. It is shown that for an LCA group 〈em〉G〈/em〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si207.svg"〉〈mi〉G〈/mi〉〈mo〉∈〈/mo〉〈mi mathvariant="fraktur"〉N〈/mi〉〈/math〉 if and only if 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si212.svg"〉〈mi〉G〈/mi〉〈mo〉≅〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msup〉〈mo〉×〈/mo〉〈mi〉H〈/mi〉〈/math〉, where 〈em〉n〈/em〉 is a non-negative integer and 〈em〉H〈/em〉 is an LCA group with an open compact subgroup 〈em〉N〈/em〉 such that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si213.svg"〉〈mi〉H〈/mi〉〈mo stretchy="false"〉/〈/mo〉〈mi〉N〈/mi〉〈mo〉∈〈/mo〉〈mi mathvariant="fraktur"〉M〈/mi〉〈/math〉. As an application of this result, we extend a well known result on discrete abelian groups to the case of LCA groups. We show that if 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mo stretchy="false"〉(〈/mo〉〈mi〉G〈/mi〉〈mo〉,〈/mo〉〈mi〉τ〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 is a non-compact LCA group and 〈em〉σ〈/em〉 is a predecessor of 〈em〉τ〈/em〉 in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si102.svg"〉〈msub〉〈mrow〉〈mi mathvariant="script"〉G〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉G〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉, then the connected component of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mo stretchy="false"〉(〈/mo〉〈mi〉G〈/mi〉〈mo〉,〈/mo〉〈mi〉τ〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 coincides with the connected component of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si274.svg"〉〈mo stretchy="false"〉(〈/mo〉〈mi〉G〈/mi〉〈mo〉,〈/mo〉〈mi〉σ〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉. It is also shown that for a non-compact LCA group 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mo stretchy="false"〉(〈/mo〉〈mi〉G〈/mi〉〈mo〉,〈/mo〉〈mi〉τ〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉, if 〈em〉σ〈/em〉 is a predecessor of 〈em〉τ〈/em〉 in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si102.svg"〉〈msub〉〈mrow〉〈mi mathvariant="script"〉G〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉G〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉, then the equality 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.svg"〉〈mi〉i〈/mi〉〈mi〉b〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉G〈/mi〉〈mo〉,〈/mo〉〈mi〉τ〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mi〉i〈/mi〉〈mi〉b〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉G〈/mi〉〈mo〉,〈/mo〉〈mi〉σ〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 holds. This partially answer a question posed in the literature.〈/p〉〈/div〉

Description: 〈p〉Publication date: 1 October 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 266〈/p〉 〈p〉Author(s): Violeta Vasilevska〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉This paper provides further investigation of the concept of shape 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi mathvariant="normal"〉m〈/mi〉〈/mrow〉〈mrow〉〈mi mathvariant="normal"〉simpl〈/mi〉〈/mrow〉〈/msub〉〈/math〉-fibrators (previously introduced by the author). The main results identify shape 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi mathvariant="normal"〉m〈/mi〉〈/mrow〉〈mrow〉〈mi mathvariant="normal"〉simpl〈/mi〉〈/mrow〉〈/msub〉〈/math〉-fibrators among direct products of Hopfian manifolds. First it is established that every closed orientable manifold homotopically determined by 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si209.svg"〉〈msub〉〈mrow〉〈mi〉π〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈/math〉 with coperfectly Hopfian group (a new class of Hopfian groups that are introduced here) is a shape 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi mathvariant="normal"〉m〈/mi〉〈/mrow〉〈mrow〉〈mi mathvariant="normal"〉simpl〈/mi〉〈/mrow〉〈/msub〉〈/math〉o-fibrator if it is a codimension-2 fibrator (Theorem 5.4). The main result (Theorem 6.2) states that the direct product of two closed orientable manifolds (of different dimension) homotopically determined by 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si209.svg"〉〈msub〉〈mrow〉〈mi〉π〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈/math〉 and with coperfectly Hopfian fundamental groups (one normally incommensurable with the other one) is a shape 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi mathvariant="normal"〉m〈/mi〉〈/mrow〉〈mrow〉〈mi mathvariant="normal"〉simpl〈/mi〉〈/mrow〉〈/msub〉〈/math〉o-fibrator, if it is a Hopfian manifold and a codimension-2 fibrator.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 September 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 265〈/p〉 〈p〉Author(s): Han Lou, Mingxing Zhang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Let 〈em〉M〈/em〉 be a simple 3-manifold, and 〈em〉F〈/em〉 be a component of ∂〈em〉M〈/em〉 of genus at least 2. Let 〈em〉α〈/em〉 and 〈em〉β〈/em〉 be separating slopes on 〈em〉F〈/em〉. Let 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mi〉M〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉α〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 (resp. 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"〉〈mi〉M〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉β〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉) be the manifold obtained by adding a 2-handle along 〈em〉α〈/em〉 (resp. 〈em〉β〈/em〉). If 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mi〉M〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉α〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.svg"〉〈mi〉M〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉β〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 are ∂-reducible, then the minimal geometric intersection number of 〈em〉α〈/em〉 and 〈em〉β〈/em〉 is at most 8.〈/p〉〈/div〉

Description: 〈p〉Publication date: 1 October 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 266〈/p〉 〈p〉Author(s): Keita Nakagane〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We show, using Mellit's recent results, that Kálmán's full twist formula for the HOMFLY polynomial can be generalized to a formula for superpolynomials in the case of positive toric braids.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 September 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 265〈/p〉 〈p〉Author(s): Jerzy Ka̧kol〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Following Christensen [11] a metrizable space 〈em〉X〈/em〉 is 〈em〉σ〈/em〉-compact if and only if 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 is analytic, i.e. it is a continuous image of the Polish space 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉N〈/mi〉〈/mrow〉〈mrow〉〈mi mathvariant="double-struck"〉N〈/mi〉〈/mrow〉〈/msup〉〈/math〉. By Michael [26] the space 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 is a continuous image of a separable metric space if and only if 〈em〉X〈/em〉 is cosmic, and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 is an 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"〉〈msub〉〈mrow〉〈mi mathvariant="normal"〉ℵ〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈/math〉-space if and only if 〈em〉X〈/em〉 is countable. We show here that, in parallel manner, Christensen-Calbrix and Michael results may be characterized as follows: For a metrizable space 〈em〉X〈/em〉 the space 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 is analytic if and only if 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 is a continuous linear image of a separable metrizable locally convex space, and a Tychonoff space 〈em〉X〈/em〉 is a submetrizable hemicompact 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"〉〈msub〉〈mrow〉〈mi〉k〈/mi〉〈/mrow〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈/msub〉〈/math〉-space if and only if 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 is a continuous linear image of a metrizable and complete separable locally convex space. Applications are provided.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 September 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 265〈/p〉 〈p〉Author(s): Dekui Peng, Wei He〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉A Hausdorff topological group 〈em〉G〈/em〉 is called 〈em〉lower continuous〈/em〉 if the topology of 〈em〉G〈/em〉 has no predecessor in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi mathvariant="script"〉G〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉G〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉. The class of lower continuous topological groups contains all closed subgroups of products of minimal abelian groups, so strictly extend the class of minimal groups. Our main concern in this paper is the study of properties of lower continuous topological groups. Similar with the case for minimal groups, we provide a lower continuity criterion: a dense subgroup 〈em〉H〈/em〉 of a Hausdorff topological abelian group 〈em〉G〈/em〉 is lower continuous if and only if 〈em〉G〈/em〉 is lower continuous and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si127.svg"〉〈mi〉S〈/mi〉〈mi〉o〈/mi〉〈mi〉c〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉G〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo〉≤〈/mo〉〈mi〉H〈/mi〉〈/math〉. It is shown that every totally lower continuous abelian group is precompact. It is also shown that for a compact abelian groups 〈em〉G〈/em〉, 〈em〉G〈/em〉 is hereditarily lower continuous if and only if 〈em〉G〈/em〉 is torsion-free.〈/p〉〈/div〉

Description: 〈p〉Publication date: 1 October 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 266〈/p〉 〈p〉Author(s): Javier Camargo, Sergio Macías, Marco Ruiz〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We study three different topics related to the Jones' set function 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mi mathvariant="script"〉T〈/mi〉〈/math〉. The first topic is idempotency; we study differences between idempotency on continua and idempotency on closed sets. The second aspect that we present are some properties about the set 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si137.svg"〉〈mi mathvariant="script"〉T〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈msup〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈mrow〉〈mi〉X〈/mi〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉)〈/mo〉〈/math〉. Particularly, we show that it is not possible to find a continuum 〈em〉X〈/em〉 such that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si137.svg"〉〈mi mathvariant="script"〉T〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈msup〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈mrow〉〈mi〉X〈/mi〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 is compact and countable; and we give a continuum 〈em〉X〈/em〉 such that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si137.svg"〉〈mi mathvariant="script"〉T〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈msup〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈mrow〉〈mi〉X〈/mi〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 is countable. Finally, the third topic is the behavior of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mi mathvariant="script"〉T〈/mi〉〈/math〉 on products. One of our main results is that the compactness of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"〉〈mi mathvariant="script"〉T〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈msup〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈mrow〉〈mi〉X〈/mi〉〈mo〉×〈/mo〉〈mi〉Y〈/mi〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 implies the local connectedness of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.svg"〉〈mi〉X〈/mi〉〈mo〉×〈/mo〉〈mi〉Y〈/mi〉〈/math〉.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 September 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 265〈/p〉 〈p〉Author(s): Wei-Feng Xuan, Yan-Kui Song〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The notion of a 〈em〉κ〈/em〉-splitting diagonal was introduced and studied by Tkachuk. In this paper, we prove that there exists a locally countable, locally compact space with an 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi〉ω〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈/math〉-splitting diagonal but no 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si12.svg"〉〈msub〉〈mrow〉〈mi〉G〈/mi〉〈/mrow〉〈mrow〉〈mi〉δ〈/mi〉〈/mrow〉〈/msub〉〈/math〉-diagonal. Using the Erdös-Radó's theorem, we also prove that every DCCC homogeneous space 〈em〉X〈/em〉 with a regular 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si12.svg"〉〈msub〉〈mrow〉〈mi〉G〈/mi〉〈/mrow〉〈mrow〉〈mi〉δ〈/mi〉〈/mrow〉〈/msub〉〈/math〉-diagonal such that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si115.svg"〉〈mi〉π〈/mi〉〈mi〉χ〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mi〉ω〈/mi〉〈/math〉 has cardinality at most 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si135.svg"〉〈mi mathvariant="fraktur"〉c〈/mi〉〈/math〉. Some new questions are also posed.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 September 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 265〈/p〉 〈p〉Author(s): S. Cobzaş〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉 〈p〉We prove versions of Ekeland, Takahashi and Caristi principles in sequentially right 〈em〉K〈/em〉-complete quasi-pseudometric spaces (meaning asymmetric pseudometric spaces), the equivalence between these principles, as well as their equivalence to the completeness of the underlying quasi-pseudometric space.〈/p〉 〈p〉The key tools are Picard sequences for some special set-valued mappings corresponding to a function 〈em〉φ〈/em〉 on a quasi-pseudometric space, allowing a unitary treatment of all these principles.〈/p〉 〈/div〉

Description: 〈p〉Publication date: 1 October 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 266〈/p〉 〈p〉Author(s): W. Xi, D. Dikranjan, M. Shlossberg, D. Toller〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉 〈p〉We study locally compact groups having all dense subgroups (locally) minimal. We call such groups 〈em〉densely (locally) minimal〈/em〉. In 1972 Prodanov proved that the infinite compact abelian groups having all subgroups minimal are precisely the groups 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi mathvariant="double-struck"〉Z〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msub〉〈/math〉 of 〈em〉p〈/em〉-adic integers. In [30], we extended Prodanov's theorem to the non-abelian case at several levels. In this paper, we focus on the densely (locally) minimal abelian groups.〈/p〉 〈p〉We prove that in case that a topological abelian group 〈em〉G〈/em〉 is either compact or connected locally compact, then 〈em〉G〈/em〉 is densely locally minimal if and only if 〈em〉G〈/em〉 either is a Lie group or has an open subgroup isomorphic to 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi mathvariant="double-struck"〉Z〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msub〉〈/math〉 for some prime 〈em〉p〈/em〉. This should be compared with the main result of [9]. Our Theorem C provides another extension of Prodanov's theorem: an infinite locally compact group is densely minimal if and only if it is isomorphic to 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi mathvariant="double-struck"〉Z〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msub〉〈/math〉. In contrast, we show that there exists a densely minimal, compact, two-step nilpotent group that neither is a Lie group nor it has an open subgroup isomorphic to 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi mathvariant="double-struck"〉Z〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msub〉〈/math〉.〈/p〉 〈/div〉

Description: 〈p〉Publication date: 15 September 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 265〈/p〉 〈p〉Author(s): Oorna Mitra, Parameswaran Sankaran〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We show that certain groups of diffeomorphisms and PL-homeomorphisms embed in the group of all quasi-isometry classes of the Euclidean spaces.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 September 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 265〈/p〉 〈p〉Author(s): Dániel T. Soukup, Paul J. Szeptycki〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉A topological space 〈em〉X〈/em〉 is strongly 〈em〉D〈/em〉 if for any neighbourhood assignment 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mo stretchy="false"〉{〈/mo〉〈msub〉〈mrow〉〈mi〉U〈/mi〉〈/mrow〉〈mrow〉〈mi〉x〈/mi〉〈/mrow〉〈/msub〉〈mo〉:〈/mo〉〈mi〉x〈/mi〉〈mo〉∈〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉}〈/mo〉〈/math〉, there is a 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈mi〉D〈/mi〉〈mo〉⊆〈/mo〉〈mi〉X〈/mi〉〈/math〉 such that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"〉〈mo stretchy="false"〉{〈/mo〉〈msub〉〈mrow〉〈mi〉U〈/mi〉〈/mrow〉〈mrow〉〈mi〉x〈/mi〉〈/mrow〉〈/msub〉〈mo〉:〈/mo〉〈mi〉x〈/mi〉〈mo〉∈〈/mo〉〈mi〉D〈/mi〉〈mo stretchy="false"〉}〈/mo〉〈/math〉 covers 〈em〉X〈/em〉 and 〈em〉D〈/em〉 is locally finite in the topology generated by 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mo stretchy="false"〉{〈/mo〉〈msub〉〈mrow〉〈mi〉U〈/mi〉〈/mrow〉〈mrow〉〈mi〉x〈/mi〉〈/mrow〉〈/msub〉〈mo〉:〈/mo〉〈mi〉x〈/mi〉〈mo〉∈〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉}〈/mo〉〈/math〉. We prove that ⋄ implies that there is an 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si308.svg"〉〈mrow〉〈mi mathvariant="normal"〉HF〈/mi〉〈msub〉〈mrow〉〈mi mathvariant="normal"〉C〈/mi〉〈/mrow〉〈mrow〉〈mi mathvariant="normal"〉w〈/mi〉〈/mrow〉〈/msub〉〈/mrow〉〈/math〉 space in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.svg"〉〈msup〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈mrow〉〈msub〉〈mrow〉〈mi〉ω〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈/mrow〉〈/msup〉〈/math〉 (hence 0-dimensional, Hausdorff and hereditarily Lindelöf) which is not strongly 〈em〉D〈/em〉. We also show that any HFC space 〈em〉X〈/em〉 is dually discrete and if additionally countable sets have Menger closure then 〈em〉X〈/em〉 is a 〈em〉D〈/em〉-space.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 September 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 265〈/p〉 〈p〉Author(s): Bingzhe Hou, Geng Tian〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, we introduce some new equivalence relations for topological dynamical systems named strong topological shift equivalence and topological shift equivalence, which are similar to the strong shift equivalence and shift equivalence for subshifts of finite type. We study the relations between the new equivalences and other equivalences such as topological conjugacy, mutually topological semi-conjugacy and canonical homeomorphism extensions being topologically conjugate. Some properties and examples are shown. In particular, mean topological dimension is an invariant for topological shift equivalence but not for mutually topologically semi-conjugate equivalence. In this topic, linear operators are also considered.〈/p〉〈/div〉

Description: 〈p〉Publication date: 1 October 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 266〈/p〉 〈p〉Author(s): Sajjad Mohammadi, Mohammad A. Asadi-Golmankhaneh〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Let 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi〉P〈/mi〉〈/mrow〉〈mrow〉〈mn〉4〈/mn〉〈mo〉,〈/mo〉〈mi〉k〈/mi〉〈/mrow〉〈/msub〉〈/math〉 be the principal 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈mi〉S〈/mi〉〈mi〉U〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mn〉4〈/mn〉〈mo stretchy="false"〉)〈/mo〉〈/math〉-bundle over 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"〉〈msup〉〈mrow〉〈mi〉S〈/mi〉〈/mrow〉〈mrow〉〈mn〉8〈/mn〉〈/mrow〉〈/msup〉〈/math〉 with Chern class 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"〉〈msub〉〈mrow〉〈mi〉c〈/mi〉〈/mrow〉〈mrow〉〈mn〉4〈/mn〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈msub〉〈mrow〉〈mi〉P〈/mi〉〈/mrow〉〈mrow〉〈mn〉4〈/mn〉〈mo〉,〈/mo〉〈mi〉k〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉)〈/mo〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mn〉6〈/mn〉〈mi〉k〈/mi〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si34.svg"〉〈msub〉〈mrow〉〈mi mathvariant="script"〉G〈/mi〉〈/mrow〉〈mrow〉〈mi〉k〈/mi〉〈/mrow〉〈/msub〉〈/math〉 be the gauge group of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi〉P〈/mi〉〈/mrow〉〈mrow〉〈mn〉4〈/mn〉〈mo〉,〈/mo〉〈mi〉k〈/mi〉〈/mrow〉〈/msub〉〈/math〉 classified by 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.svg"〉〈mi〉k〈/mi〉〈msup〉〈mrow〉〈mi〉ε〈/mi〉〈/mrow〉〈mrow〉〈mo〉′〈/mo〉〈/mrow〉〈/msup〉〈/math〉, where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si7.svg"〉〈msup〉〈mrow〉〈mi〉ε〈/mi〉〈/mrow〉〈mrow〉〈mo〉′〈/mo〉〈/mrow〉〈/msup〉〈/math〉 a generator of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si8.svg"〉〈msub〉〈mrow〉〈mi〉π〈/mi〉〈/mrow〉〈mrow〉〈mn〉8〈/mn〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉B〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉S〈/mi〉〈mi〉U〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mn〉4〈/mn〉〈mo stretchy="false"〉)〈/mo〉〈mo stretchy="false"〉)〈/mo〉〈mo stretchy="false"〉)〈/mo〉〈mo〉≅〈/mo〉〈mi mathvariant="double-struck"〉Z〈/mi〉〈/math〉. In this article we partially classify the homotopy types of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si34.svg"〉〈msub〉〈mrow〉〈mi mathvariant="script"〉G〈/mi〉〈/mrow〉〈mrow〉〈mi〉k〈/mi〉〈/mrow〉〈/msub〉〈/math〉 by showing that if there is a homotopy equivalence 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si190.svg"〉〈msub〉〈mrow〉〈mi mathvariant="script"〉G〈/mi〉〈/mrow〉〈mrow〉〈mi〉k〈/mi〉〈/mrow〉〈/msub〉〈mo〉≃〈/mo〉〈msub〉〈mrow〉〈mi mathvariant="script"〉G〈/mi〉〈/mrow〉〈mrow〉〈msup〉〈mrow〉〈mi〉k〈/mi〉〈/mrow〉〈mrow〉〈mo〉′〈/mo〉〈/mrow〉〈/msup〉〈/mrow〉〈/msub〉〈/math〉 then 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.svg"〉〈mo stretchy="false"〉(〈/mo〉〈mn〉420〈/mn〉〈mo〉,〈/mo〉〈mi〉k〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mo stretchy="false"〉(〈/mo〉〈mn〉420〈/mn〉〈mo〉,〈/mo〉〈msup〉〈mrow〉〈mi〉k〈/mi〉〈/mrow〉〈mrow〉〈mo〉′〈/mo〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 and if 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"〉〈mo stretchy="false"〉(〈/mo〉〈mn〉3360〈/mn〉〈mo〉,〈/mo〉〈mi〉k〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 is equal to 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si12.svg"〉〈mo stretchy="false"〉(〈/mo〉〈mn〉3360〈/mn〉〈mo〉,〈/mo〉〈msup〉〈mrow〉〈mi〉k〈/mi〉〈/mrow〉〈mrow〉〈mo〉′〈/mo〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 then 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si190.svg"〉〈msub〉〈mrow〉〈mi mathvariant="script"〉G〈/mi〉〈/mrow〉〈mrow〉〈mi〉k〈/mi〉〈/mrow〉〈/msub〉〈mo〉≃〈/mo〉〈msub〉〈mrow〉〈mi mathvariant="script"〉G〈/mi〉〈/mrow〉〈mrow〉〈msup〉〈mrow〉〈mi〉k〈/mi〉〈/mrow〉〈mrow〉〈mo〉′〈/mo〉〈/mrow〉〈/msup〉〈/mrow〉〈/msub〉〈/math〉.〈/p〉〈/div〉

Description: 〈p〉Publication date: 1 October 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 266〈/p〉 〈p〉Author(s): A. Bartoš, J. Bobok, J. van Mill, P. Pyrih, B. Vejnar〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We introduce the notion of compactifiable classes – these are classes of metrizable compact spaces that can be up to homeomorphic copies “disjointly combined” into one metrizable compact space. This is witnessed by so-called compact composition of the class. Analogously, we consider Polishable classes and Polish compositions. The question of compactifiability or Polishability of a class is related to hyperspaces. Strongly compactifiable and strongly Polishable classes may be characterized by the existence of a corresponding family in the hyperspace of all metrizable compacta. We systematically study the introduced notions – we give several characterizations, consider preservation under various constructions, and raise several questions.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 29 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications〈/p〉 〈p〉Author(s): Carolina Medina, Gelasio Salazar〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Let 〈em〉L〈/em〉 be a fixed link. Given a link diagram 〈em〉D〈/em〉, is there a sequence of crossing exchanges and smoothings on 〈em〉D〈/em〉 that yields a diagram of 〈em〉L〈/em〉? We approach this problem from the computational complexity point of view. It follows from work by Endo, Itoh, and Taniyama that if 〈em〉L〈/em〉 is a prime link with crossing number at most 5, then there is an algorithm that answers this question in polynomial time. We show that the same holds for all torus links 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉T〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈mo〉,〈/mo〉〈mi〉m〈/mi〉〈/mrow〉〈/msub〉〈/math〉 and all twist knots.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 27 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications〈/p〉 〈p〉Author(s): Akira Koyama, Vesko Valov〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We provide some properties and characterizations of homologically 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mi〉U〈/mi〉〈msup〉〈mrow〉〈mi〉V〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msup〉〈/math〉-maps and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈mi〉l〈/mi〉〈msubsup〉〈mrow〉〈mi〉c〈/mi〉〈/mrow〉〈mrow〉〈mi〉G〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msubsup〉〈/math〉-spaces. We show that there is a parallel between recently introduced by Cauty [3] algebraic 〈em〉ANR〈/em〉's and homologically 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈mi〉l〈/mi〉〈msubsup〉〈mrow〉〈mi〉c〈/mi〉〈/mrow〉〈mrow〉〈mi〉G〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msubsup〉〈/math〉-metric spaces, and this parallel is similar to the parallel between ordinary 〈em〉ANR〈/em〉's and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.gif" overflow="scroll"〉〈mi〉L〈/mi〉〈msup〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msup〉〈/math〉-metric spaces. We also show that there is a similarity between the properties of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.gif" overflow="scroll"〉〈mi〉L〈/mi〉〈msup〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msup〉〈/math〉-spaces and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈mi〉l〈/mi〉〈msubsup〉〈mrow〉〈mi〉c〈/mi〉〈/mrow〉〈mrow〉〈mi〉G〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msubsup〉〈/math〉-spaces. Some open questions are raised.〈/p〉〈/div〉

Description: 〈p〉Publication date: 1 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 254〈/p〉 〈p〉Author(s): Henry Adams, Joshua Mirth〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Given a sample 〈em〉Y〈/em〉 from an unknown manifold 〈em〉X〈/em〉 embedded in Euclidean space, it is possible to recover the homology groups of 〈em〉X〈/em〉 by building a Vietoris–Rips or Čech simplicial complex on top of the vertex set 〈em〉Y〈/em〉. However, these simplicial complexes need not inherit the metric structure of the manifold, in particular when 〈em〉Y〈/em〉 is infinite. Indeed, a simplicial complex is not even metrizable if it is not locally finite. We instead consider metric thickenings, called the 〈em〉Vietoris–Rips〈/em〉 and 〈em〉Čech thickenings〈/em〉, which are equipped with the 1-Wasserstein metric in place of the simplicial complex topology. We show that for Euclidean subsets 〈em〉X〈/em〉 with positive reach, the thickenings satisfy metric analogues of Hausmann's theorem and the nerve lemma (the metric Vietoris–Rips and Čech thickenings of 〈em〉X〈/em〉 are homotopy equivalent to 〈em〉X〈/em〉 for scale parameters less than the reach). To our knowledge this is the first version of Hausmann's theorem for Vietoris–Rips constructions on entire Euclidean submanifolds (as opposed to Riemannian manifolds), and our result also extends to non-manifold shapes (as not all sets of positive reach are manifolds). In contrast to Hausmann's original proof, our homotopy equivalence is a deformation retraction, is realized by canonical maps in both directions, and furthermore can be proven to be a homotopy equivalence via simple linear homotopies from the map compositions to the corresponding identity maps.〈/p〉〈/div〉

Description: 〈p〉Publication date: 1 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 254〈/p〉 〈p〉Author(s): Verónica Martínez-de-la-Vega, Jorge M. Martínez-Montejano, Christopher Mouron〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper we show that if 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mi〉h〈/mi〉〈mo〉:〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉→〈/mo〉〈mi〉X〈/mi〉〈/math〉 is a mixing homeomorphism on a 〈em〉G〈/em〉-like continuum, then 〈em〉X〈/em〉 must be indecomposable and if 〈em〉X〈/em〉 is finitely cyclic, then 〈em〉X〈/em〉 must be 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si169.gif" overflow="scroll"〉〈mfrac〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/mfrac〉〈/math〉-indecomposable for some natural number 〈em〉n〈/em〉. Furthermore, we give an example of a mixing homeomorphism on a hereditary decomposable tree-like continuum.〈/p〉〈/div〉

Description: 〈p〉Publication date: 1 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 254〈/p〉 〈p〉Author(s): Fortunata Aurora Basile, Maddalena Bonanzinga, Nathan Carlson, Jack Porter〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper we extend the theory of H-closed extensions of Hausdorff spaces to a class of non-Hausdorff spaces, defined in [2], called 〈em〉n〈/em〉-Hausdorff spaces. The notion of H-closed is generalized to an 〈em〉n〈/em〉-H-closed space. Known construction for Hausdorff spaces 〈em〉X〈/em〉, such as the Katětov H-closed extension 〈em〉κX〈/em〉, are generalized to a maximal 〈em〉n〈/em〉-H-closed extension denoted by 〈em〉n〈/em〉-〈em〉κX〈/em〉.〈/p〉〈/div〉

Description: 〈p〉Publication date: 1 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 254〈/p〉 〈p〉Author(s): Akram Alishahi, Nathan Dowlin〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We show that the page at which the Lee spectral sequence collapses gives a bound on the unknotting number, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mi〉u〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉K〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉. In particular, for knots with 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈mi〉u〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉K〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo〉≤〈/mo〉〈mn〉2〈/mn〉〈/math〉, we show that the Lee spectral sequence must collapse at the 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉E〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈/math〉 page. An immediate corollary is that the Knight Move Conjecture is true when 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈mi〉u〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉K〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo〉≤〈/mo〉〈mn〉2〈/mn〉〈/math〉.〈/p〉〈/div〉

Description: 〈p〉Publication date: 1 November 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 267〈/p〉 〈p〉Author(s): Urtzi Buijs, José M. Moreno-Fernández〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We provide two criteria for establishing the non-formality of a differential graded Lie algebra in terms of higher Whitehead brackets, which are the Lie analogues of the Massey products of a differential graded associative algebra. We also show that formality of a differential graded Lie algebra is not equivalent to the collapse of its associated Quillen spectral sequence. Finally, we use 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mo〉∞〈/mo〉〈/mrow〉〈/msub〉〈/math〉 algebras and Quillen's formulation of rational homotopy theory to recover and improve a classical theorem for detecting higher Whitehead products in Sullivan minimal models, and give some applications.〈/p〉〈/div〉

Description: 〈p〉Publication date: 1 November 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 267〈/p〉 〈p〉Author(s): Michael Albanese, Aleksandar Milivojević〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We record an answer to the question “In which dimensions is the connected sum of two closed almost complex manifolds necessarily an almost complex manifold?”. In the process of doing so, we are naturally led to ask “For which values of 〈em〉ℓ〈/em〉 is the connected sum of 〈em〉ℓ〈/em〉 closed almost complex manifolds necessarily an almost complex manifold?”. We answer this question, along with its non-compact analogue, using obstruction theory and Yang's results on the existence of almost complex structures on 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mo stretchy="false"〉(〈/mo〉〈mi〉n〈/mi〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈mn〉1〈/mn〉〈mo stretchy="false"〉)〈/mo〉〈/math〉-connected 2〈em〉n〈/em〉-manifolds. Finally, we partially extend Datta and Subramanian's result on the nonexistence of almost complex structures on products of two even spheres to rational homology spheres by using the index of the twisted 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈msup〉〈mrow〉〈mtext〉spin〈/mtext〉〈/mrow〉〈mrow〉〈mi〉c〈/mi〉〈/mrow〉〈/msup〉〈/math〉 Dirac operator.〈/p〉〈/div〉

Description: 〈p〉Publication date: 1 November 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 267〈/p〉 〈p〉Author(s): Henry Jose Gullo Mercado, Leandro Fiorini Aurichi〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Let 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo〉,〈/mo〉〈mi〉τ〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 be a topological space and let 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈mi mathvariant="script"〉F〈/mi〉〈/math〉 be the family of all subsets of 〈em〉X〈/em〉 that satisfy a given topological property 〈em〉P〈/em〉 (invariant under homeomorphisms). If we add new open sets to the topology and if 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"〉〈msup〉〈mrow〉〈mi mathvariant="script"〉F〈/mi〉〈/mrow〉〈mrow〉〈mo〉′〈/mo〉〈/mrow〉〈/msup〉〈/math〉 is the family of all subsets of the new space which satisfy the property 〈em〉P〈/em〉, we can have 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"〉〈mi mathvariant="script"〉F〈/mi〉〈mo〉≠〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="script"〉F〈/mi〉〈/mrow〉〈mrow〉〈mo〉′〈/mo〉〈/mrow〉〈/msup〉〈/math〉. If this is always the case, we say that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo〉,〈/mo〉〈mi〉τ〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 is maximal with respect to the family 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈mi mathvariant="script"〉F〈/mi〉〈/math〉. We show here some characterizations of maximal spaces with respect to the family of discrete subsets.〈/p〉〈/div〉

Description: 〈p〉Publication date: 1 November 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 267〈/p〉 〈p〉Author(s): Cerene Rathilal, Dharmanand Baboolal, Paranjothi Pillay〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Given a locally connected metric frame 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mo stretchy="false"〉(〈/mo〉〈mi〉L〈/mi〉〈mo〉,〈/mo〉〈mi〉d〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 a new compatible metric diameter on 〈em〉L〈/em〉 is shown to exist, which in the spatial case corresponds to a metric due to Kelley [8] having the property that all spherical neighbourhoods of a point are connected and have property S. Our main result is that if a locally connected metric frame has property S then its top element can be written as finite join of arbitrarily small connected elements each having property S.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 25 September 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications〈/p〉 〈p〉Author(s): Maria Trnková〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper we address some problems concerning an approximate Dirichlet domain. We show that under some assumptions an approximate Dirichlet domain can work equally well as an exact Dirichlet domain. In particular, we consider a problem of tiling a hyperbolic ball with copies of the Dirichlet domain. This problem arises in the construction of the length spectrum algorithm which is implemented by the computer program SnapPea. Our result explains the empirical fact that the program works surprisingly well despite it does not use exact data. Also we demonstrate a rigorous verification whether two words of the fundamental group of a hyperbolic 3-manifold are the same or not.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 25 September 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications〈/p〉 〈p〉Author(s): Zava Nicolò〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We consider two categories: the category 〈strong〉Coarse〈/strong〉 of coarse spaces and bornologous maps and its quotient category 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mi mathvariant="bold"〉Coarse〈/mi〉〈msub〉〈mrow〉〈mo stretchy="false"〉/〈/mo〉〈/mrow〉〈mrow〉〈mo〉∼〈/mo〉〈/mrow〉〈/msub〉〈/math〉, where ∼ is the closeness relation. This paper tackles the problem of their wellpoweredness and cowellpoweredness. In particular, we show that all the epireflective subcategories of 〈strong〉Coarse〈/strong〉 are cowellpowered, using a complete characterisation of closure operators of 〈strong〉Coarse〈/strong〉, while 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mi mathvariant="bold"〉Coarse〈/mi〉〈msub〉〈mrow〉〈mo stretchy="false"〉/〈/mo〉〈/mrow〉〈mrow〉〈mo〉∼〈/mo〉〈/mrow〉〈/msub〉〈/math〉 is both wellpowered and cowellpowered. Moreover, we prove that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mi mathvariant="bold"〉Coarse〈/mi〉〈msub〉〈mrow〉〈mo stretchy="false"〉/〈/mo〉〈/mrow〉〈mrow〉〈mo〉∼〈/mo〉〈/mrow〉〈/msub〉〈/math〉 has neither equalizers nor pullbacks of subobjects, although it has arbitrary products.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 25 September 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications〈/p〉 〈p〉Author(s): Andrzej Kucharski, Sławomir Turek〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We introduce a new class of 〈em〉ϰ〈/em〉-metrizable spaces, namely countably 〈em〉ϰ〈/em〉-metrizable spaces. We show that the class of all 〈em〉ϰ〈/em〉-metrizable spaces is a proper subclass of countably 〈em〉ϰ〈/em〉-metrizable spaces. On the other hand, for pseudocompact spaces the new class coincides with 〈em〉ϰ〈/em〉-metrizable spaces. We prove a generalization of Chigogidze's result that the Čech-Stone compactification of a pseudocompact countably 〈em〉ϰ〈/em〉-metrizable space is 〈em〉ϰ〈/em〉-metrizable.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 25 September 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications〈/p〉 〈p〉Author(s): Angelo Bella, Alan Dow〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Few observations on a paper of Arhangel'skiĭ and Buzyakova led us to consider Rančin's problem. The main result here is the construction under ⋄ of a compact c-sequential space that is not sequential.〈/p〉〈/div〉

Description: 〈p〉Publication date: 1 November 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 267〈/p〉 〈p〉Author(s): 〈/p〉

Description: 〈p〉Publication date: 1 December 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 268〈/p〉 〈p〉Author(s): Angelo Bella〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The inequality 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mo stretchy="false"〉|〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉|〈/mo〉〈mo〉≤〈/mo〉〈msup〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈mrow〉〈mi〉χ〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/mrow〉〈/msup〉〈/math〉 has been proved to be true for both the class of Lindelöf spaces (Arhangel'skii, 1969 [1]) and that of 〈em〉H〈/em〉-closed spaces (Dow and Porter, 1982 [6]), by different arguments. We present a common weakening of the Lindelöf and 〈em〉H〈/em〉-closed properties which allows us to give a unified proof of these two theorems.〈/p〉〈/div〉

Description: 〈p〉Publication date: 1 December 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 268〈/p〉 〈p〉Author(s): Jaume Llibre, Víctor F. Sirvent〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In the present article we give sufficient conditions for 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msup〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mo〉∞〈/mo〉〈/mrow〉〈/msup〉〈/math〉 self-maps on some connected compact manifolds in order to have positive entropy. The conditions are given in terms of the Lefschetz numbers of the iterates of the map and/or its Lefschetz zeta function. We consider the cases where the manifold is a compact orientable and non-orientable surface, the 〈em〉n〈/em〉-dimensional torus, the product of 〈em〉n〈/em〉 spheres of dimension 〈em〉ℓ〈/em〉 and the product of spheres of different dimensions.〈/p〉〈/div〉

Description: 〈p〉Publication date: 1 December 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 268〈/p〉 〈p〉Author(s): Lev Bukovský, Alexander V. Osipov〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉 〈p〉Let 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msubsup〉〈mrow〉〈mi mathvariant="normal"〉USC〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈mrow〉〈mo〉⋆〈/mo〉〈/mrow〉〈/msubsup〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 be the topological space of real upper semicontinuous bounded functions defined on 〈em〉X〈/em〉 with the subspace topology of the product topology on 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈mmultiscripts〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mprescripts〉〈/mprescripts〉〈none〉〈/none〉〈mrow〉〈mi〉X〈/mi〉〈/mrow〉〈/mmultiscripts〉〈/math〉. 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"〉〈msup〉〈mrow〉〈mover accent="true"〉〈mrow〉〈mi mathvariant="normal"〉Φ〈/mi〉〈/mrow〉〈mrow〉〈mo stretchy="false"〉˜〈/mo〉〈/mrow〉〈/mover〉〈/mrow〉〈mrow〉〈mo stretchy="false"〉↑〈/mo〉〈/mrow〉〈/msup〉〈mo〉,〈/mo〉〈msup〉〈mrow〉〈mover accent="true"〉〈mrow〉〈mi mathvariant="normal"〉Ψ〈/mi〉〈/mrow〉〈mrow〉〈mo stretchy="false"〉˜〈/mo〉〈/mrow〉〈/mover〉〈/mrow〉〈mrow〉〈mo stretchy="false"〉↑〈/mo〉〈/mrow〉〈/msup〉〈/math〉 are the sets of all upper sequentially dense, upper dense or pointwise dense subsets of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msubsup〉〈mrow〉〈mi mathvariant="normal"〉USC〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈mrow〉〈mo〉⋆〈/mo〉〈/mrow〉〈/msubsup〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉, respectively. We prove several equivalent assertions to that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msubsup〉〈mrow〉〈mi mathvariant="normal"〉USC〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈mrow〉〈mo〉⋆〈/mo〉〈/mrow〉〈/msubsup〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 satisfies the selection principles S〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"〉〈msub〉〈mrow〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈msup〉〈mrow〉〈mover accent="true"〉〈mrow〉〈mi mathvariant="normal"〉Φ〈/mi〉〈/mrow〉〈mrow〉〈mo stretchy="false"〉˜〈/mo〉〈/mrow〉〈/mover〉〈/mrow〉〈mrow〉〈mo stretchy="false"〉↑〈/mo〉〈/mrow〉〈/msup〉〈mo〉,〈/mo〉〈msup〉〈mrow〉〈mover accent="true"〉〈mrow〉〈mi mathvariant="normal"〉Ψ〈/mi〉〈/mrow〉〈mrow〉〈mo stretchy="false"〉˜〈/mo〉〈/mrow〉〈/mover〉〈/mrow〉〈mrow〉〈mo stretchy="false"〉↑〈/mo〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉)〈/mo〉〈/math〉, including a condition on the topological space 〈em〉X〈/em〉.〈/p〉 〈p〉We prove similar results for the topological space 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si140.svg"〉〈msubsup〉〈mrow〉〈mi mathvariant="normal"〉C〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈mrow〉〈mo〉⋆〈/mo〉〈/mrow〉〈/msubsup〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 of continuous bounded functions.〈/p〉 〈p〉Similar results hold true for the selection principles S〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.svg"〉〈msub〉〈mrow〉〈/mrow〉〈mrow〉〈mi〉f〈/mi〉〈mi〉i〈/mi〉〈mi〉n〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈msup〉〈mrow〉〈mover accent="true"〉〈mrow〉〈mi mathvariant="normal"〉Φ〈/mi〉〈/mrow〉〈mrow〉〈mo stretchy="false"〉˜〈/mo〉〈/mrow〉〈/mover〉〈/mrow〉〈mrow〉〈mo stretchy="false"〉↑〈/mo〉〈/mrow〉〈/msup〉〈mo〉,〈/mo〉〈msup〉〈mrow〉〈mover accent="true"〉〈mrow〉〈mi mathvariant="normal"〉Ψ〈/mi〉〈/mrow〉〈mrow〉〈mo stretchy="false"〉˜〈/mo〉〈/mrow〉〈/mover〉〈/mrow〉〈mrow〉〈mo stretchy="false"〉↑〈/mo〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉)〈/mo〉〈/math〉.〈/p〉 〈/div〉

Description: 〈p〉Publication date: 1 February 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 270〈/p〉 〈p〉Author(s): Volodymyr Mykhaylyuk, Vadym Myronyk〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We analyze compactness-like properties of sets in partial metric spaces and obtain the equivalence of several forms of the compactness for partial metric spaces. Moreover, we give a negative answer to a question from [8] on the existence of a bounded complete partial metric on a complete partial metric space.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 7 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications〈/p〉 〈p〉Author(s): Salvador Hernández, F. Javier Trigos-Arrieta〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We look at the Bohr topology of maximally almost periodic groups (MAP, for short). Among other results, we investigate when a Hausdorff precompact abelian group 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mo stretchy="false"〉(〈/mo〉〈mi〉G〈/mi〉〈mo〉,〈/mo〉〈mi mathvariant="normal"〉w〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 is the Bohr reflection of a locally compact abelian group. Necessary and sufficient conditions are established in terms of the inner properties of the topology w. As an application, an example of a MAP group 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈mo stretchy="false"〉(〈/mo〉〈mi〉G〈/mi〉〈mo〉,〈/mo〉〈mi〉t〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 is given such that every closed, metrizable subgroup 〈em〉N〈/em〉 of 〈em〉bG〈/em〉 with 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.gif" overflow="scroll"〉〈mi〉N〈/mi〉〈mo〉∩〈/mo〉〈mi〉G〈/mi〉〈mo〉=〈/mo〉〈mo stretchy="false"〉{〈/mo〉〈mn〉0〈/mn〉〈mo stretchy="false"〉}〈/mo〉〈/math〉 preserves compactness but 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈mo stretchy="false"〉(〈/mo〉〈mi〉G〈/mi〉〈mo〉,〈/mo〉〈mi〉t〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 does not strongly respect compactness. Thereby, we respond to Questions 4.1 and 4.3 in [6].〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 April 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 257〈/p〉 〈p〉Author(s): Daron Anderson, Paul Bankston〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉 〈p〉We import into continuum theory the notion of 〈em〉extreme point of a convex set〈/em〉 from the theory of topological vector spaces. We explore how extreme points relate to other established types of “edge point” of a continuum; for example we prove that extreme points are always shore points, and that any extreme point is also non-block if the continuum is either decomposable or irreducible (in particular, metrizable).〈/p〉 〈p〉In addition we discuss some continuum-theoretic analogues of the celebrated Krein-Milman theorem.〈/p〉 〈/div〉

Description: 〈p〉Publication date: 15 May 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 258〈/p〉 〈p〉Author(s): Mohammad Javaheri〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We show that if a homeomorphism of a separable locally compact metric space has a unique fixed point that is attracting or repelling, then its corresponding composition operator is cyclic. On the real line, we show that a composition operator is cyclic if and only if its symbol has at most one fixed point. Other results on the real line and circle are also discussed.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 April 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 257〈/p〉 〈p〉Author(s): Taketo Shirane〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The splitting number is effective to distinguish the embedded topology of plane curves, and it is not determined by the fundamental group of the complement of the plane curve. In this paper, we give a generalization of the splitting number, called the 〈em〉splitting graph〈/em〉. By using the splitting graph, we classify the embedded topology of plane curves consisting of one smooth curve and non-concurrent three lines, called 〈em〉Artal arrangements〈/em〉.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 May 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 258〈/p〉 〈p〉Author(s): Xin Liu, Chuan Liu, Shou Lin〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉 〈p〉Based on the notions of T. Banakh's strict Pytkeev networks and A.V. Arhangel'skiı̌'s sensor families, strict Pytkeev networks with sensors are introduced in this paper. A family 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mi mathvariant="script"〉P〈/mi〉〈/math〉 of subsets of a topological space 〈em〉X〈/em〉 is called a 〈em〉strict Pytkeev network with sensors〈/em〉 (abbr. an 〈em〉sp-network〈/em〉) if, for each 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈mi〉x〈/mi〉〈mo〉∈〈/mo〉〈mi〉U〈/mi〉〈mo〉∩〈/mo〉〈mover accent="true"〉〈mrow〉〈mi〉A〈/mi〉〈/mrow〉〈mo〉‾〈/mo〉〈/mover〉〈/math〉 with 〈em〉U〈/em〉 open and 〈em〉A〈/em〉 subset in 〈em〉X〈/em〉, there is a set 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.gif" overflow="scroll"〉〈mi〉P〈/mi〉〈mo〉∈〈/mo〉〈mi mathvariant="script"〉P〈/mi〉〈/math〉 such that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.gif" overflow="scroll"〉〈mi〉x〈/mi〉〈mo〉∈〈/mo〉〈mi〉P〈/mi〉〈mo〉⊂〈/mo〉〈mi〉U〈/mi〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si49.gif" overflow="scroll"〉〈mi〉x〈/mi〉〈mo〉∈〈/mo〉〈mover accent="true"〉〈mrow〉〈mi〉P〈/mi〉〈mo〉∩〈/mo〉〈mi〉A〈/mi〉〈/mrow〉〈mo〉‾〈/mo〉〈/mover〉〈/math〉. In present paper, we discuss certain relationship and operations among spaces defined by special Pytkeev networks, study spaces with a point-countable 〈em〉sp〈/em〉-network and spaces with a 〈em〉σ〈/em〉-closure-preserving 〈em〉sp〈/em〉-network, and detect some applications of 〈em〉sp〈/em〉-networks in topological groups.〈/p〉 〈p〉The following results are obtained: (1) Every 〈em〉sp〈/em〉-network is preserved by a continuous pseudo-open mapping. (2) Every 〈em〉k〈/em〉-space with a point-countable 〈em〉sp〈/em〉-network coincides with a continuous pseudo-open 〈em〉s〈/em〉-image of a metric space. (3) Every regular feebly compact space with a point-countable 〈em〉sp〈/em〉-network has a point-countable base. (4) A regular space has a countable 〈em〉sp〈/em〉-network if and only if it is separable and has a point-countable 〈em〉sp〈/em〉-network. (5) A topological space is stratifiable if and only if it is a regular space with a 〈em〉σ〈/em〉-closure-preserving 〈em〉sp〈/em〉-network. (6) A regular space with a 〈em〉σ〈/em〉-locally finite 〈em〉sp〈/em〉-network has a 〈em〉σ〈/em〉-discrete 〈em〉sp〈/em〉-network. (7) A topological group is metrizable if it has countable 〈em〉sp〈/em〉-character. (8) There is a non-Fréchet-Urysohn sequential topological group with a countable strict Pytkeev network, which give a negative answer to a question posed by A.V. Arhangel'skiı̌ [1].〈/p〉 〈/div〉

Description: 〈p〉Publication date: 15 May 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 258〈/p〉 〈p〉Author(s): Mostafa Abedi〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Let 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mi mathvariant="script"〉R〈/mi〉〈mi〉L〈/mi〉〈/math〉 be the ring of continuous real-valued functions on a completely regular frame 〈em〉L〈/em〉. We study the class of prime ideals 〈em〉P〈/em〉 of the ring 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mi mathvariant="script"〉R〈/mi〉〈mi〉L〈/mi〉〈/math〉 determined by the condition: 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈mi mathvariant="script"〉R〈/mi〉〈mi〉L〈/mi〉〈mo stretchy="false"〉/〈/mo〉〈mi〉P〈/mi〉〈/math〉 is a real-closed ring. We give some necessary and sufficient frame-theoretic conditions for an ordered integral domain of the form 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈mi mathvariant="script"〉R〈/mi〉〈mi〉L〈/mi〉〈mo stretchy="false"〉/〈/mo〉〈mi〉P〈/mi〉〈/math〉 to be a real-closed ring, when 〈em〉P〈/em〉 is a prime 〈em〉z〈/em〉-ideal (〈em〉d〈/em〉-ideal) of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mi mathvariant="script"〉R〈/mi〉〈mi〉L〈/mi〉〈/math〉. A completely regular frame 〈em〉L〈/em〉 is called an 〈em〉SV〈/em〉-frame if for every prime ideal 〈em〉P〈/em〉 of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mi mathvariant="script"〉R〈/mi〉〈mi〉L〈/mi〉〈/math〉, the ordered integral domain 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈mi mathvariant="script"〉R〈/mi〉〈mi〉L〈/mi〉〈mo stretchy="false"〉/〈/mo〉〈mi〉P〈/mi〉〈/math〉 is a real-closed ring. It is shown that every 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si235.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mo〉⁎〈/mo〉〈/mrow〉〈/msup〉〈/math〉-quotient of an 〈em〉SV〈/em〉-frame is an 〈em〉SV〈/em〉-frame. We also show that open quotients ↓〈em〉c〈/em〉 in an 〈em〉SV〈/em〉-frame are 〈em〉SV〈/em〉-frames for all cozero elements 〈em〉c〈/em〉. Larson [22] has given a topological characterization of compact 〈em〉SV〈/em〉-spaces. By extending this characterization to frames, we show that the compactness limitation can really be relaxed, even in spaces, and so strengthen Larson's result.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 20 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications〈/p〉 〈p〉Author(s): Chi-Keung Ng〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉 〈p〉We introduce the notion of coarse metric. Every coarse metric induces a coarse structure on the underlying set. Conversely, we observe that all coarse spaces come from a particular type of coarse metric in a unique way. In the case when the coarse structure 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mi mathvariant="script"〉E〈/mi〉〈/math〉 on a set 〈em〉X〈/em〉 is defined by a coarse metric that takes values in a meet-complete totally ordered set, we define the associated Hausdorff coarse metric on the set 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi mathvariant="script"〉P〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 of non-empty subsets of 〈em〉X〈/em〉 and show that it induces the Hausdorff coarse structure on 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi mathvariant="script"〉P〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉.〈/p〉 〈p〉On the other hand, we define the notion of pseudo uniform metric. Each pseudo uniform metric induces a uniform structure on the underlying space. In the reverse direction, we show that a uniform structure 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.gif" overflow="scroll"〉〈mi mathvariant="script"〉U〈/mi〉〈/math〉 on a set 〈em〉X〈/em〉 is induced by a map 〈em〉d〈/em〉 from 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.gif" overflow="scroll"〉〈mi〉X〈/mi〉〈mo〉×〈/mo〉〈mi〉X〈/mi〉〈/math〉 to a partially ordered set (with no requirement on 〈em〉d〈/em〉) if and only if 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.gif" overflow="scroll"〉〈mi mathvariant="script"〉U〈/mi〉〈/math〉 admits a base 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si109.gif" overflow="scroll"〉〈mi mathvariant="script"〉B〈/mi〉〈/math〉 such that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si17.gif" overflow="scroll"〉〈mi mathvariant="script"〉B〈/mi〉〈mo〉∪〈/mo〉〈mo stretchy="false"〉{〈/mo〉〈mo〉⋂〈/mo〉〈mi mathvariant="script"〉U〈/mi〉〈mo stretchy="false"〉}〈/mo〉〈/math〉 is closed under arbitrary intersections. In this case, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.gif" overflow="scroll"〉〈mi mathvariant="script"〉U〈/mi〉〈/math〉 is actually defined by a pseudo uniform metric. We also show that a uniform structures 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.gif" overflow="scroll"〉〈mi mathvariant="script"〉U〈/mi〉〈/math〉 comes from a pseudo uniform metric that takes values in a totally ordered set if and only if 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.gif" overflow="scroll"〉〈mi mathvariant="script"〉U〈/mi〉〈/math〉 admits a totally ordered base.〈/p〉 〈p〉Finally, a valuation ring will produce an example of a coarse and pseudo uniform metric that take values in a totally ordered set.〈/p〉 〈/div〉

Description: 〈p〉Publication date: 15 May 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 258〈/p〉 〈p〉Author(s): Héctor Barge, José M.R. Sanjurjo〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper we study continuous parametrized families of dissipative flows, which are those flows having a global attractor. The main motivation for this study comes from the observation that, in general, global attractors are not robust, in the sense that small perturbations of the flow can destroy their globality. We give a necessary and sufficient condition for a global attractor to be continued to a global attractor. We also study, using shape theoretical methods and the Conley index, the bifurcation global to non-global.〈/p〉〈/div〉

Description: 〈p〉Publication date: 1 September 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 264〈/p〉 〈p〉Author(s): Javier Camargo, Mayer Palacios, Hugo Villanueva〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, we study when a strongly freely decomposable mapping is almost monotone. Also, we define the notion of 〈em〉i〈/em〉-unicoherent continuum and present some properties and examples of this class of continua.〈/p〉〈/div〉

Description: 〈p〉Publication date: 1 September 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 264〈/p〉 〈p〉Author(s): Kyeonghui Lee, Young Ho Im, Sera Kim〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We introduce a family of polynomial invariants for flat virtual knots which extend the polynomial introduced by Kauffman and Richter, and we give several properties and examples.〈/p〉〈/div〉

Description: 〈p〉Publication date: 1 September 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 264〈/p〉 〈p〉Author(s): Kengo Kawamura〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉An embedded/immersed surface-knot is a closed and connected surface embedded/immersed in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mn〉4〈/mn〉〈/mrow〉〈/msup〉〈/math〉, respectively. The triple point number 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈mi〉t〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉F〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 of an embedded/immersed surface-knot 〈em〉F〈/em〉 is the minimum number of triple points required for a diagram of 〈em〉F〈/em〉. Satoh proved that (i) there does not exist an embedded surface-knot 〈em〉F〈/em〉 such that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.svg"〉〈mi〉t〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉F〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mn〉1〈/mn〉〈/math〉, and (ii) there does not exist an embedded 2-knot 〈em〉F〈/em〉 such that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"〉〈mi〉t〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉F〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mn〉2〈/mn〉〈mtext〉 or 〈/mtext〉〈mn〉3〈/mn〉〈/math〉. In this paper, we prove similar results for immersed surface-knots with some conditions.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 June 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 260〈/p〉 〈p〉Author(s): Juan Carlos Martínez, Lajos Soukup〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉 〈p〉We prove the following consistency result for cardinal sequences of length 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mo〉〈〈/mo〉〈msub〉〈mrow〉〈mi〉ω〈/mi〉〈/mrow〉〈mrow〉〈mn〉3〈/mn〉〈/mrow〉〈/msub〉〈/math〉: if GCH holds and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si115.gif" overflow="scroll"〉〈mi〉λ〈/mi〉〈mo〉≥〈/mo〉〈msub〉〈mrow〉〈mi〉ω〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈/math〉 is a regular cardinal, then in some cardinal-preserving generic extension 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si116.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈mrow〉〈mi〉ω〈/mi〉〈/mrow〉〈/msup〉〈mo〉=〈/mo〉〈mi〉λ〈/mi〉〈/math〉 and for every ordinal 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si117.gif" overflow="scroll"〉〈mi〉η〈/mi〉〈mo〉〈〈/mo〉〈msub〉〈mrow〉〈mi〉ω〈/mi〉〈/mrow〉〈mrow〉〈mn〉3〈/mn〉〈/mrow〉〈/msub〉〈/math〉 and every sequence 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si118.gif" overflow="scroll"〉〈mi〉f〈/mi〉〈mo〉=〈/mo〉〈mo stretchy="false"〉〈〈/mo〉〈msub〉〈mrow〉〈mi〉κ〈/mi〉〈/mrow〉〈mrow〉〈mi〉α〈/mi〉〈/mrow〉〈/msub〉〈mo〉:〈/mo〉〈mi〉α〈/mi〉〈mo〉〈〈/mo〉〈mi〉η〈/mi〉〈mo stretchy="false"〉〉〈/mo〉〈/math〉 of infinite cardinals with 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si119.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉κ〈/mi〉〈/mrow〉〈mrow〉〈mi〉α〈/mi〉〈/mrow〉〈/msub〉〈mo〉≤〈/mo〉〈mi〉λ〈/mi〉〈/math〉 for 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si105.gif" overflow="scroll"〉〈mi〉α〈/mi〉〈mo〉〈〈/mo〉〈mi〉η〈/mi〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si120.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉κ〈/mi〉〈/mrow〉〈mrow〉〈mi〉α〈/mi〉〈/mrow〉〈/msub〉〈mo〉=〈/mo〉〈mi〉ω〈/mi〉〈/math〉 if 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si68.gif" overflow="scroll"〉〈mtext〉cf〈/mtext〉〈mo stretchy="false"〉(〈/mo〉〈mi〉α〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo〉=〈/mo〉〈msub〉〈mrow〉〈mi〉ω〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈/math〉, we have that 〈em〉f〈/em〉 is the cardinal sequence of some LCS space.〈/p〉 〈p〉Also, we prove that for every specific uncountable cardinal 〈em〉λ〈/em〉 it is relatively consistent with ZFC that for every 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.gif" overflow="scroll"〉〈mi〉α〈/mi〉〈mo〉,〈/mo〉〈mi〉β〈/mi〉〈mo〉〈〈/mo〉〈msub〉〈mrow〉〈mi〉ω〈/mi〉〈/mrow〉〈mrow〉〈mn〉3〈/mn〉〈/mrow〉〈/msub〉〈/math〉 with 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.gif" overflow="scroll"〉〈mtext〉cf〈/mtext〉〈mo stretchy="false"〉(〈/mo〉〈mi〉α〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo〉〈〈/mo〉〈msub〉〈mrow〉〈mi〉ω〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈/math〉 there is an LCS space 〈em〉Z〈/em〉 such that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si12.gif" overflow="scroll"〉〈mtext〉CS〈/mtext〉〈mo stretchy="false"〉(〈/mo〉〈mi〉Z〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo〉=〈/mo〉〈msub〉〈mrow〉〈mo stretchy="false"〉〈〈/mo〉〈mi〉ω〈/mi〉〈mo stretchy="false"〉〉〈/mo〉〈/mrow〉〈mrow〉〈mi〉α〈/mi〉〈/mrow〉〈/msub〉〈msup〉〈mrow〉〈/mrow〉〈mrow〉〈mo〉⌢〈/mo〉〈/mrow〉〈/msup〉〈mspace width="0.2em"〉〈/mspace〉〈msub〉〈mrow〉〈mo stretchy="false"〉〈〈/mo〉〈mi〉λ〈/mi〉〈mo stretchy="false"〉〉〈/mo〉〈/mrow〉〈mrow〉〈mi〉β〈/mi〉〈/mrow〉〈/msub〉〈/math〉.〈/p〉 〈/div〉

Description: 〈p〉Publication date: 15 June 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 260〈/p〉 〈p〉Author(s): Sergey V. Ludkovsky〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The article is devoted to microbundles over topological rings. Their structure, homomorphisms, automorphisms and extensions are studied. Moreover, compactifications and inverse spectra of microbundles over topological rings are investigated.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 August 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 263〈/p〉 〈p〉Author(s): Yan-Kui Song, Wei-Feng Xuan〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Bal and Kočinac in [2] introduced and studied the class of selectively star-ccc spaces. A space 〈em〉X〈/em〉 is called 〈em〉selectively star-ccc〈/em〉 if for every open cover 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mi mathvariant="script"〉U〈/mi〉〈/math〉 of 〈em〉X〈/em〉 and for every sequence 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈mo stretchy="false"〉(〈/mo〉〈msub〉〈mrow〉〈mi mathvariant="script"〉A〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msub〉〈mo〉:〈/mo〉〈mi〉n〈/mi〉〈mo〉∈〈/mo〉〈mi〉ω〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 of maximal pairwise disjoint open families in 〈em〉X〈/em〉, there exists a sequence 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si23.svg"〉〈mo stretchy="false"〉(〈/mo〉〈msub〉〈mrow〉〈mi〉A〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msub〉〈mo〉:〈/mo〉〈mi〉n〈/mi〉〈mo〉∈〈/mo〉〈mi〉ω〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 such that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"〉〈msub〉〈mrow〉〈mi〉A〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msub〉〈mo〉∈〈/mo〉〈msub〉〈mrow〉〈mi mathvariant="script"〉A〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msub〉〈/math〉 for every 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.svg"〉〈mi〉n〈/mi〉〈mo〉∈〈/mo〉〈mi〉ω〈/mi〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si25.svg"〉〈mi mathvariant="normal"〉St〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈msub〉〈mrow〉〈mo〉⋃〈/mo〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈mo〉∈〈/mo〉〈mi〉ω〈/mi〉〈/mrow〉〈/msub〉〈msub〉〈mrow〉〈mi〉A〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msub〉〈mo〉,〈/mo〉〈mi mathvariant="script"〉U〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mi〉X〈/mi〉〈/math〉. In this paper, we show that there exists a Tychonoff selectively 2-star-ccc space which is neither strongly star Lindelöf nor selectively star-ccc, which gives a positive answer to a question of Bal and Kočinac [2]. Under 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si26.svg"〉〈msup〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈mrow〉〈msub〉〈mrow〉〈mi mathvariant="normal"〉ℵ〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈/mrow〉〈/msup〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈msup〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈mrow〉〈msub〉〈mrow〉〈mi mathvariant="normal"〉ℵ〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈/mrow〉〈/msup〉〈/math〉, we even provide a normal example of a selectively 2-star-ccc space which is neither strongly star Lindelöf nor selectively star-ccc. Finally, we prove that every open 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si8.svg"〉〈msub〉〈mrow〉〈mi〉F〈/mi〉〈/mrow〉〈mrow〉〈mi〉σ〈/mi〉〈/mrow〉〈/msub〉〈/math〉-subset of a selectively star-ccc space is selectively star-ccc. Some new questions are also posed.〈/p〉〈/div〉

Description: 〈p〉Publication date: 1 September 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 264〈/p〉 〈p〉Author(s): Noboru Ito, Migiwa Sakurai〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Vassiliev introduced filtered invariants of knots using an unknotting operation, called crossing changes. Goussarov, Polyak, and Viro introduced other filtered invariants of virtual knots, which order is called GPV-order, using an unknotting operation, called virtualization. We defined other filtered invariants, which order is called 〈em〉F〈/em〉-order, of virtual knots using an unknotting operation, called forbidden moves. In this paper, we show that the set of virtual knot invariants of 〈em〉F〈/em〉-order 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mo〉≤〈/mo〉〈mi〉n〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉+〈/mo〉〈mn〉1〈/mn〉〈/math〉 is strictly stronger than that of 〈em〉F〈/em〉-order ≤〈em〉n〈/em〉 and that of GPV-order 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈mo〉≤〈/mo〉〈mn〉2〈/mn〉〈mi〉n〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉+〈/mo〉〈mn〉1〈/mn〉〈/math〉. To obtain the result, we show that the set of virtual knot invariants of 〈em〉F〈/em〉-order ≤〈em〉n〈/em〉 contains every Goussarov-Polyak-Viro invariant of GPV-order 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈mo〉≤〈/mo〉〈mn〉2〈/mn〉〈mi〉n〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉+〈/mo〉〈mn〉1〈/mn〉〈/math〉, which implies that the set of virtual knot invariants of 〈em〉F〈/em〉-order is a complete invariant of classical and virtual knots.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 August 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 263〈/p〉 〈p〉Author(s): Elisa Hartmann〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉This paper presents a new version of boundary on coarse spaces. The space of ends functor maps coarse metric spaces to uniform topological spaces and coarse maps to uniformly continuous maps.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 28 February 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications〈/p〉 〈p〉Author(s): Salvador Hernández, Dieter Remus, F. Javier Trigos-Arrieta〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The important rôle that W. W. Comfort played in the study of the Bohr topology is described.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 May 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 258〈/p〉 〈p〉Author(s): De Kui Peng, Wei He〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉 〈p〉Let 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mi mathvariant="fraktur"〉M〈/mi〉〈/math〉 be a family of Hausdorff topological groups. A Hausdorff topological group 〈em〉P〈/em〉 is called 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mi mathvariant="fraktur"〉M〈/mi〉〈/math〉-〈em〉cancellable〈/em〉 if for any two topological groups 〈em〉G〈/em〉 and 〈em〉H〈/em〉 in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mi mathvariant="fraktur"〉M〈/mi〉〈/math〉, the product group 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈mi〉G〈/mi〉〈mo〉×〈/mo〉〈mi〉P〈/mi〉〈/math〉 is topologically isomorphic to 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.gif" overflow="scroll"〉〈mi〉H〈/mi〉〈mo〉×〈/mo〉〈mi〉P〈/mi〉〈/math〉 if and only if 〈em〉G〈/em〉 and 〈em〉H〈/em〉 are topologically isomorphic. If 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mi mathvariant="fraktur"〉M〈/mi〉〈/math〉 is the class of all Hausdorff topological groups, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mi mathvariant="fraktur"〉M〈/mi〉〈/math〉-〈em〉cancellable〈/em〉 topological groups are called 〈em〉cancellable topological groups〈/em〉.〈/p〉 〈p〉We show in this paper that every topological group in a family 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si139.gif" overflow="scroll"〉〈mi mathvariant="fraktur"〉N〈/mi〉〈/math〉 of topological groups is cancellable. In particular, the additive group 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si150.gif" overflow="scroll"〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/math〉 of reals and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si151.gif" overflow="scroll"〉〈mi mathvariant="double-struck"〉Q〈/mi〉〈/math〉 of rationals endowed with the usual topologies are both cancellable. We also show that every topological group in a family 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mi mathvariant="fraktur"〉M〈/mi〉〈/math〉 of topological groups is cancellable. In particular, every topologically simple non-abelian group is cancellable. It is also proved that every topologically simple Hopfian or co-Hopfian topological group is cancellable.〈/p〉 〈/div〉

Description: 〈p〉Publication date: Available online 28 February 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications〈/p〉 〈p〉Author(s): Neil Hindman〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉I survey the publications of Wis Comfort whose main subject was either the Stone-Čech compactification of a completely regular Hausdorff space or the set of ultrafilters on a given set. These areas are tied together by the fact that if 〈em〉X〈/em〉 is a discrete space, its Stone-Čech compactification can be viewed as the set of ultrafilters on 〈em〉X〈/em〉.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 28 February 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications〈/p〉 〈p〉Author(s): W.W. Comfort, A.W. Hager, J. van Mill〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉A Tychonoff space 〈em〉X〈/em〉 will be called strongly bicompactly condensable (SBC) if there is a set 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mi mathvariant="script"〉S〈/mi〉〈/math〉 of compact Hausdorff topologies on the set 〈em〉X〈/em〉 whose supremum in the lattice of topologies is the original topology. Such an 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mi mathvariant="script"〉S〈/mi〉〈/math〉 determines a compactification 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈mi〉K〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi mathvariant="script"〉S〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 of 〈em〉X〈/em〉. We examine which compactifications of an SBC 〈em〉X〈/em〉 arise in this way: For some 〈em〉X〈/em〉, all do, and for others, some and not all; For some 〈em〉X〈/em〉, 〈em〉βX〈/em〉 does, and for others, does not.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 28 February 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications〈/p〉 〈p〉Author(s): K.E. Jordan, K. Marinelli, T.J. Peters, J.A. Roulier, P. Zaffetti〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Knots are prevalent mathematical models in molecular algorithms, requiring isotopically equivalent piecewise linear (PL) approximations for visualization and analyses. Local topological properties of Bézier curves are shown to support efficient isotopic approximations, inclusive of illustrative computational experiments. For this memorial issue, author T.J. Peters notes that ‘computational topology’ was not discussed during his mentorship by W.W. Comfort, but existence of this contemporary subfield is testimony to the breadth of Comfort's mathematical legacy.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 28 February 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications〈/p〉 〈p〉Author(s): Artur Hideyuki Tomita〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉 〈p〉We show that if 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mi〉κ〈/mi〉〈mo〉≤〈/mo〉〈mi〉ω〈/mi〉〈/math〉 and there exists a group topology without non-trivial convergent sequences on an Abelian group 〈em〉H〈/em〉 such that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mi〉H〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msup〉〈/math〉 is countably compact for each 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.gif" overflow="scroll"〉〈mi〉n〈/mi〉〈mo〉〈〈/mo〉〈mi〉κ〈/mi〉〈/math〉 then there exists a topological group 〈em〉G〈/em〉 such that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mi〉G〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msup〉〈/math〉 is countably compact for each 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.gif" overflow="scroll"〉〈mi〉n〈/mi〉〈mo〉〈〈/mo〉〈mi〉κ〈/mi〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mi〉G〈/mi〉〈/mrow〉〈mrow〉〈mi〉κ〈/mi〉〈/mrow〉〈/msup〉〈/math〉 is not countably compact. If in addition 〈em〉H〈/em〉 is torsion, then the result above holds for 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.gif" overflow="scroll"〉〈mi〉κ〈/mi〉〈mo〉=〈/mo〉〈msub〉〈mrow〉〈mi〉ω〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈/math〉. Combining with other results in the literature, we show that:〈/p〉 〈p〉〈em〉a〈/em〉) Assuming 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si321.gif" overflow="scroll"〉〈mi mathvariant="fraktur"〉c〈/mi〉〈/math〉 incomparable selective ultrafilters, for each 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si8.gif" overflow="scroll"〉〈mi〉n〈/mi〉〈mo〉∈〈/mo〉〈mi〉ω〈/mi〉〈/math〉, there exists a group topology on the free Abelian group 〈em〉G〈/em〉 such that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mi〉G〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msup〉〈/math〉 is countably compact and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mi〉G〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈mo〉+〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈/msup〉〈/math〉 is not countably compact. (It was already know for 〈em〉ω〈/em〉).〈/p〉 〈p〉〈em〉b〈/em〉) If 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.gif" overflow="scroll"〉〈mi〉κ〈/mi〉〈mo〉∈〈/mo〉〈mi〉ω〈/mi〉〈mo〉∪〈/mo〉〈mo stretchy="false"〉{〈/mo〉〈mi〉ω〈/mi〉〈mo stretchy="false"〉}〈/mo〉〈mo〉∪〈/mo〉〈mo stretchy="false"〉{〈/mo〉〈msub〉〈mrow〉〈mi〉ω〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉}〈/mo〉〈/math〉, there exists in ZFC a topological group 〈em〉G〈/em〉 such that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mi〉G〈/mi〉〈/mrow〉〈mrow〉〈mi〉γ〈/mi〉〈/mrow〉〈/msup〉〈/math〉 is countably compact for each cardinal 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si12.gif" overflow="scroll"〉〈mi〉γ〈/mi〉〈mo〉〈〈/mo〉〈mi〉κ〈/mi〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mi〉G〈/mi〉〈/mrow〉〈mrow〉〈mi〉κ〈/mi〉〈/mrow〉〈/msup〉〈/math〉 is not countably compact.〈/p〉 〈/div〉

Description: 〈p〉Publication date: Available online 28 February 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications〈/p〉 〈p〉Author(s): Dikran Dikranjan〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉This is a survey on the contribution of W. Wistar Comfort to the theory of topological groups through the looking glass of “dense subgroups of topological groups”. It turns out that under this umbrella one can find an amazing variety of results, as well as problems posed by Comfort that gave a significant input in the progress in the field of topological groups in the last five decades.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 28 February 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications〈/p〉 〈p〉Author(s): Taras Banakh, Serhii Bardyla〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉 〈p〉We study the interplay between three weak topologies on a topological semilattice 〈em〉X〈/em〉: the weak〈sup〉∘〈/sup〉 topology 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈msubsup〉〈mrow〉〈mi〉w〈/mi〉〈/mrow〉〈mrow〉〈mi〉X〈/mi〉〈/mrow〉〈mrow〉〈mo〉∘〈/mo〉〈/mrow〉〈/msubsup〉〈/math〉 (generated by the base consisting of open subsemilattices of 〈em〉X〈/em〉), the weak〈sup〉•〈/sup〉 topology 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si131.gif" overflow="scroll"〉〈msubsup〉〈mrow〉〈mi〉w〈/mi〉〈/mrow〉〈mrow〉〈mi〉X〈/mi〉〈/mrow〉〈mrow〉〈mo〉•〈/mo〉〈/mrow〉〈/msubsup〉〈/math〉 (generated by the subbase consisting of complements to closed subsemilattices), and the 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si159.gif" overflow="scroll"〉〈mi mathvariant="double-struck"〉I〈/mi〉〈/math〉-weak topology 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si156.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉w〈/mi〉〈/mrow〉〈mrow〉〈mi〉X〈/mi〉〈/mrow〉〈/msub〉〈/math〉 (which is the weakest topology in which all continuous homomorphisms 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.gif" overflow="scroll"〉〈mi〉h〈/mi〉〈mo〉:〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉→〈/mo〉〈mo stretchy="false"〉[〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mn〉1〈/mn〉〈mo stretchy="false"〉]〈/mo〉〈/math〉 remain continuous). Also we study the interplay between the weak topologies 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si131.gif" overflow="scroll"〉〈msub〉〈mrow〉〈msup〉〈mrow〉〈mi〉w〈/mi〉〈/mrow〉〈mrow〉〈mo〉•〈/mo〉〈/mrow〉〈/msup〉〈/mrow〉〈mrow〉〈mi〉X〈/mi〉〈/mrow〉〈/msub〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈msubsup〉〈mrow〉〈mi〉w〈/mi〉〈/mrow〉〈mrow〉〈mi〉X〈/mi〉〈/mrow〉〈mrow〉〈mo〉∘〈/mo〉〈/mrow〉〈/msubsup〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si156.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉w〈/mi〉〈/mrow〉〈mrow〉〈mi〉X〈/mi〉〈/mrow〉〈/msub〉〈/math〉 of a topological semilattice 〈em〉X〈/em〉 and some intrinsic topologies, determined by the order structure of the semilattice.〈/p〉 〈p〉We prove that the weak〈sup〉•〈/sup〉 topology 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si131.gif" overflow="scroll"〉〈msubsup〉〈mrow〉〈mi〉w〈/mi〉〈/mrow〉〈mrow〉〈mi〉X〈/mi〉〈/mrow〉〈mrow〉〈mo〉•〈/mo〉〈/mrow〉〈/msubsup〉〈/math〉 on a Hausdorff semitopological semilattice 〈em〉X〈/em〉 is compact if and only if 〈em〉X〈/em〉 is chain-compact in the sense that each closed chain in 〈em〉X〈/em〉 is compact. For a compact Hausdorff topological semilattice 〈em〉X〈/em〉 with topology 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si366.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉τ〈/mi〉〈/mrow〉〈mrow〉〈mi〉X〈/mi〉〈/mrow〉〈/msub〉〈/math〉 we prove that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si8.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉τ〈/mi〉〈/mrow〉〈mrow〉〈mi〉X〈/mi〉〈/mrow〉〈/msub〉〈mo〉=〈/mo〉〈msub〉〈mrow〉〈mi〉w〈/mi〉〈/mrow〉〈mrow〉〈mi〉X〈/mi〉〈/mrow〉〈/msub〉〈/math〉 iff 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉τ〈/mi〉〈/mrow〉〈mrow〉〈mi〉X〈/mi〉〈/mrow〉〈/msub〉〈mo〉=〈/mo〉〈msubsup〉〈mrow〉〈mi〉w〈/mi〉〈/mrow〉〈mrow〉〈mi〉X〈/mi〉〈/mrow〉〈mrow〉〈mo〉•〈/mo〉〈/mrow〉〈/msubsup〉〈/math〉 iff 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉τ〈/mi〉〈/mrow〉〈mrow〉〈mi〉X〈/mi〉〈/mrow〉〈/msub〉〈mo〉=〈/mo〉〈msubsup〉〈mrow〉〈mi〉w〈/mi〉〈/mrow〉〈mrow〉〈mi〉X〈/mi〉〈/mrow〉〈mrow〉〈mo〉∘〈/mo〉〈/mrow〉〈/msubsup〉〈/math〉.〈/p〉 〈/div〉

Description: 〈p〉Publication date: Available online 28 February 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications〈/p〉 〈p〉Author(s): Dmitri Shakhmatov, Víctor Hugo Yañez〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We prove that every free group 〈em〉G〈/em〉 with infinitely many generators admits a Hausdorff group topology 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mi mathvariant="script"〉T〈/mi〉〈/math〉 with the following property: for every 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mi mathvariant="script"〉T〈/mi〉〈/math〉-open neighbourhood 〈em〉U〈/em〉 of the identity of 〈em〉G〈/em〉, each element 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈mi〉g〈/mi〉〈mo〉∈〈/mo〉〈mi〉G〈/mi〉〈/math〉 can be represented as a product 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.gif" overflow="scroll"〉〈mi〉g〈/mi〉〈mo〉=〈/mo〉〈msub〉〈mrow〉〈mi〉g〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈msub〉〈mrow〉〈mi〉g〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈mo〉…〈/mo〉〈msub〉〈mrow〉〈mi〉g〈/mi〉〈/mrow〉〈mrow〉〈mi〉k〈/mi〉〈/mrow〉〈/msub〉〈/math〉, where 〈em〉k〈/em〉 is a positive integer (depending on 〈em〉g〈/em〉) and the cyclic group generated by each 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉g〈/mi〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈/mrow〉〈/msub〉〈/math〉 is contained in 〈em〉U〈/em〉. In particular, 〈em〉G〈/em〉 admits a Hausdorff group topology with the small subgroup generating property of Gould. This provides a positive answer to a question of Comfort and Gould in the case of free groups with infinitely many generators. The case of free groups with finitely many generators remains open.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 28 February 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications〈/p〉 〈p〉Author(s): Ivan S. Gotchev〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉 〈p〉For a Urysohn space 〈em〉X〈/em〉 we define the 〈em〉regular diagonal degree〈/em〉 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mover accent="true"〉〈mrow〉〈mi mathvariant="normal"〉Δ〈/mi〉〈/mrow〉〈mo〉‾〈/mo〉〈/mover〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 of 〈em〉X〈/em〉 to be the minimal infinite cardinal 〈em〉κ〈/em〉 such that 〈em〉X〈/em〉 has a regular 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si108.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉G〈/mi〉〈/mrow〉〈mrow〉〈mi〉κ〈/mi〉〈/mrow〉〈/msub〉〈/math〉-diagonal i.e. there is a family 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.gif" overflow="scroll"〉〈mo stretchy="false"〉(〈/mo〉〈msub〉〈mrow〉〈mi〉U〈/mi〉〈/mrow〉〈mrow〉〈mi〉η〈/mi〉〈/mrow〉〈/msub〉〈mo〉:〈/mo〉〈mi〉η〈/mi〉〈mo〉〈〈/mo〉〈mi〉κ〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 of open neighborhoods of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi mathvariant="normal"〉Δ〈/mi〉〈/mrow〉〈mrow〉〈mi〉X〈/mi〉〈/mrow〉〈/msub〉〈mo〉=〈/mo〉〈mo stretchy="false"〉{〈/mo〉〈mo stretchy="false"〉(〈/mo〉〈mi〉x〈/mi〉〈mo〉,〈/mo〉〈mi〉x〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo〉∈〈/mo〉〈msup〉〈mrow〉〈mi〉X〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mo〉:〈/mo〉〈mi〉x〈/mi〉〈mo〉∈〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉}〈/mo〉〈/math〉 in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si155.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mi〉X〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈/math〉 such that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi mathvariant="normal"〉Δ〈/mi〉〈/mrow〉〈mrow〉〈mi〉X〈/mi〉〈/mrow〉〈/msub〉〈mo〉=〈/mo〉〈msub〉〈mrow〉〈mo〉⋂〈/mo〉〈/mrow〉〈mrow〉〈mi〉η〈/mi〉〈mo〉〈〈/mo〉〈mi〉κ〈/mi〉〈/mrow〉〈/msub〉〈msub〉〈mrow〉〈mover accent="true"〉〈mrow〉〈mi〉U〈/mi〉〈/mrow〉〈mo〉‾〈/mo〉〈/mover〉〈/mrow〉〈mrow〉〈mi〉η〈/mi〉〈/mrow〉〈/msub〉〈/math〉.〈/p〉 〈p〉In this paper we show that if 〈em〉X〈/em〉 is a Urysohn space, then: (1) 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si170.gif" overflow="scroll"〉〈mo stretchy="false"〉|〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉|〈/mo〉〈mo〉≤〈/mo〉〈msup〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈mrow〉〈mi〉c〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo〉⋅〈/mo〉〈mover accent="true"〉〈mrow〉〈mi mathvariant="normal"〉Δ〈/mi〉〈/mrow〉〈mo〉‾〈/mo〉〈/mover〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/mrow〉〈/msup〉〈/math〉; (2) 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si222.gif" overflow="scroll"〉〈mo stretchy="false"〉|〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉|〈/mo〉〈mo〉≤〈/mo〉〈msup〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈mrow〉〈mover accent="true"〉〈mrow〉〈mi mathvariant="normal"〉Δ〈/mi〉〈/mrow〉〈mo〉‾〈/mo〉〈/mover〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo〉⋅〈/mo〉〈msup〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈mrow〉〈mi〉w〈/mi〉〈mi〉L〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/mrow〉〈/msup〉〈/mrow〉〈/msup〉〈/math〉; (3) 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.gif" overflow="scroll"〉〈mo stretchy="false"〉|〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉|〈/mo〉〈mo〉≤〈/mo〉〈mi〉w〈/mi〉〈mi〉L〈/mi〉〈msup〉〈mrow〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/mrow〉〈mrow〉〈mover accent="true"〉〈mrow〉〈mi mathvariant="normal"〉Δ〈/mi〉〈/mrow〉〈mo〉‾〈/mo〉〈/mover〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo〉⋅〈/mo〉〈mi〉χ〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/mrow〉〈/msup〉〈/math〉; and (4) 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.gif" overflow="scroll"〉〈mo stretchy="false"〉|〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉|〈/mo〉〈mo〉≤〈/mo〉〈mi〉a〈/mi〉〈mi〉L〈/mi〉〈msup〉〈mrow〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/mrow〉〈mrow〉〈mover accent="true"〉〈mrow〉〈mi mathvariant="normal"〉Δ〈/mi〉〈/mrow〉〈mo〉‾〈/mo〉〈/mover〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/mrow〉〈/msup〉〈/math〉; where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.gif" overflow="scroll"〉〈mi〉χ〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si12.gif" overflow="scroll"〉〈mi〉c〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.gif" overflow="scroll"〉〈mi〉w〈/mi〉〈mi〉L〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si14.gif" overflow="scroll"〉〈mi〉a〈/mi〉〈mi〉L〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 are respectively the character, the cellularity, the weak Lindelöf number and the almost Lindelöf number of 〈em〉X〈/em〉.〈/p〉 〈p〉The first inequality extends to the uncountable case Buzyakova's result that the cardinality of a ccc-space with a regular 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉G〈/mi〉〈/mrow〉〈mrow〉〈mi〉δ〈/mi〉〈/mrow〉〈/msub〉〈/math〉-diagonal does not exceed 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈mrow〉〈mi〉ω〈/mi〉〈/mrow〉〈/msup〉〈/math〉. It follows from (2) that every weakly Lindelöf space with a regular 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉G〈/mi〉〈/mrow〉〈mrow〉〈mi〉δ〈/mi〉〈/mrow〉〈/msub〉〈/math〉-diagonal has cardinality at most 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si17.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈mrow〉〈msup〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈mrow〉〈mi〉ω〈/mi〉〈/mrow〉〈/msup〉〈/mrow〉〈/msup〉〈/math〉.〈/p〉 〈p〉Inequality (3) implies that if 〈em〉X〈/em〉 is a space with a regular 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉G〈/mi〉〈/mrow〉〈mrow〉〈mi〉δ〈/mi〉〈/mrow〉〈/msub〉〈/math〉-diagonal, then 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si18.gif" overflow="scroll"〉〈mo stretchy="false"〉|〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉|〈/mo〉〈mo〉≤〈/mo〉〈mi〉w〈/mi〉〈mi〉L〈/mi〉〈msup〉〈mrow〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/mrow〉〈mrow〉〈mi〉χ〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/mrow〉〈/msup〉〈/math〉. This improves significantly Bell, Ginsburg and Woods inequality 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si19.gif" overflow="scroll"〉〈mo stretchy="false"〉|〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉|〈/mo〉〈mo〉≤〈/mo〉〈msup〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈mrow〉〈mi〉χ〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mi〉w〈/mi〉〈mi〉L〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/mrow〉〈/msup〉〈/math〉 for the class of normal spaces with regular 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉G〈/mi〉〈/mrow〉〈mrow〉〈mi〉δ〈/mi〉〈/mrow〉〈/msub〉〈/math〉-diagonals. In particular (3) shows that the cardinality of every first countable space with a regular 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉G〈/mi〉〈/mrow〉〈mrow〉〈mi〉δ〈/mi〉〈/mrow〉〈/msub〉〈/math〉-diagonal does not exceed 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si20.gif" overflow="scroll"〉〈mi〉w〈/mi〉〈mi〉L〈/mi〉〈msup〉〈mrow〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/mrow〉〈mrow〉〈mi〉ω〈/mi〉〈/mrow〉〈/msup〉〈/math〉.〈/p〉 〈p〉For the class of spaces with regular 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉G〈/mi〉〈/mrow〉〈mrow〉〈mi〉δ〈/mi〉〈/mrow〉〈/msub〉〈/math〉-diagonals (4) improves Bella and Cammaroto inequality 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si21.gif" overflow="scroll"〉〈mo stretchy="false"〉|〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉|〈/mo〉〈mo〉≤〈/mo〉〈msup〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈mrow〉〈mi〉χ〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo〉⋅〈/mo〉〈mi〉a〈/mi〉〈mi〉L〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/mrow〉〈/msup〉〈/math〉, which is valid for all Urysohn spaces. Also, it follows from (4) that the cardinality of every space with a regular 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉G〈/mi〉〈/mrow〉〈mrow〉〈mi〉δ〈/mi〉〈/mrow〉〈/msub〉〈/math〉-diagonal does not exceed 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si22.gif" overflow="scroll"〉〈mi〉a〈/mi〉〈mi〉L〈/mi〉〈msup〉〈mrow〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/mrow〉〈mrow〉〈mi〉ω〈/mi〉〈/mrow〉〈/msup〉〈/math〉.〈/p〉 〈/div〉

Description: 〈p〉Publication date: Available online 27 February 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications〈/p〉 〈p〉Author(s): A.W. Hager, J. van Mill〈/p〉

Description: 〈p〉Publication date: Available online 27 February 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications〈/p〉 〈p〉Author(s): Ken Ross〈/p〉

Description: 〈p〉Publication date: Available online 27 February 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications〈/p〉 〈p〉Author(s): Saak S. Gabriyelyan, Sidney A. Morris〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The Separable Quotient Problem of Banach and Mazur asks if every infinite-dimensional Banach space has an infinite-dimensional separable quotient Banach space. It has remained unsolved for 85 years but has been answered in the affirmative for special cases such as reflexive Banach spaces. An affirmative answer to the Separable Quotient Problem would obviously imply that every infinite-dimensional Banach space has a quotient topological group which is separable, metrizable, and infinite-dimensional in the sense of topology. In this paper it is proved that every infinite-dimensional Banach space has as a quotient group the separable metrizable infinite-dimensional topological group, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉T〈/mi〉〈/mrow〉〈mrow〉〈mi〉ω〈/mi〉〈/mrow〉〈/msup〉〈/math〉, where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈mi mathvariant="double-struck"〉T〈/mi〉〈/math〉 denotes the compact unit circle group. Indeed it is shown that every locally convex space, which has a subspace which is an infinite-dimensional Fréchet space, has 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉T〈/mi〉〈/mrow〉〈mrow〉〈mi〉ω〈/mi〉〈/mrow〉〈/msup〉〈/math〉 as a quotient group.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 28 February 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications〈/p〉 〈p〉Author(s): István Juhász, Lajos Soukup, Zoltán Szentmiklóssy〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉 〈p〉Given a topological property 〈em〉P〈/em〉, we say that the space 〈em〉X〈/em〉 is 〈em〉P〈/em〉-〈em〉generated〈/em〉 if for any subset 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mi〉A〈/mi〉〈mo〉⊂〈/mo〉〈mi〉X〈/mi〉〈/math〉 that is not open in 〈em〉X〈/em〉 there is a subspace 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈mi〉Y〈/mi〉〈mo〉⊂〈/mo〉〈mi〉X〈/mi〉〈/math〉 with property 〈em〉P〈/em〉 such that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.gif" overflow="scroll"〉〈mi〉A〈/mi〉〈mo〉∩〈/mo〉〈mi〉Y〈/mi〉〈/math〉 is not open in 〈em〉Y〈/em〉. (Of course, in this definition we could replace “open” with “closed”.)〈/p〉 〈p〉In this paper we prove the following two results:〈/p〉 〈dl〉 〈dt〉(1)〈/dt〉 〈dd〉〈p〉Every Lindelöf-generated regular space 〈em〉X〈/em〉 satisfying 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si14.gif" overflow="scroll"〉〈mo stretchy="false"〉|〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉|〈/mo〉〈mo〉=〈/mo〉〈mi mathvariant="normal"〉Δ〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo〉=〈/mo〉〈msub〉〈mrow〉〈mi〉ω〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈/math〉 is 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉ω〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈/math〉-resolvable.〈/p〉〈/dd〉 〈dt〉(2)〈/dt〉 〈dd〉〈p〉Any (countable extent)-generated regular space 〈em〉X〈/em〉 satisfying 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si126.gif" overflow="scroll"〉〈mi mathvariant="normal"〉Δ〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo〉〉〈/mo〉〈mi〉ω〈/mi〉〈/math〉 is 〈em〉ω〈/em〉-resolvable.〈/p〉〈/dd〉 〈/dl〉 〈p〉These are significant strengthenings of our earlier results from [9] which can be obtained from (1) and (2) by simply omitting the “-generated” part. Moreover, the second result improves a recent result of Filatova and Osipov from [4] which states that Lindelöf-generated regular spaces of uncountable dispersion character are 2-resolvable.〈/p〉 〈/div〉

Description: 〈p〉Publication date: Available online 28 February 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications〈/p〉 〈p〉Author(s): A.V. Arhangel'skii〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The general question considered below is: how the properties of a homogeneous compact space 〈em〉X〈/em〉 are influenced by the fact that 〈em〉X〈/em〉 contains a “nice” closed subspace 〈em〉F〈/em〉 which is also “nicely” located in 〈em〉X〈/em〉?. A typical result says that if the tightness of 〈em〉F〈/em〉 is countable, and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mi〉X〈/mi〉〈mo〉∖〈/mo〉〈mi〉F〈/mi〉〈/math〉 is metalindelöf, then the tightness of 〈em〉X〈/em〉 is also countable. A question posed by Oleg Pavlov is answered. Some applications of the results obtained to perfect maps are given. The concept of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉G〈/mi〉〈/mrow〉〈mrow〉〈mi〉δ〈/mi〉〈/mrow〉〈/msub〉〈/math〉-homogeneous space is introduced, and it is shown, in particular, that if 〈em〉X〈/em〉 is a 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉G〈/mi〉〈/mrow〉〈mrow〉〈mi〉δ〈/mi〉〈/mrow〉〈/msub〉〈/math〉-homogeneous compact space with countable tightness, then the weight of 〈em〉X〈/em〉 does not exceed 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si101.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈mrow〉〈mi〉ω〈/mi〉〈/mrow〉〈/msup〉〈/math〉 (Corollary 3.8).〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 28 February 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications〈/p〉 〈p〉Author(s): Zhiqiang Xiao, Iván Sánchez, Mikhail Tkachenko〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We continue to study the properties of the class 〈em〉CSS〈/em〉 of topological groups in which all closed subgroups are separable. One of the main problems here is whether the class 〈em〉CSS〈/em〉 is finitely productive or not. We solve the problem in the negative and present three examples with different combinations of additional properties: (1) there exist in 〈em〉ZFC〈/em〉 precompact Abelian groups 〈em〉H〈/em〉 and 〈em〉K〈/em〉 in 〈em〉CSS〈/em〉 such that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mi〉H〈/mi〉〈mo〉×〈/mo〉〈mi〉K〈/mi〉〈/math〉 is not in 〈em〉CSS〈/em〉; (2) under the assumption of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si12.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈mrow〉〈msub〉〈mrow〉〈mi〉ω〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈/mrow〉〈/msup〉〈mo〉=〈/mo〉〈msup〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈mrow〉〈mi〉ω〈/mi〉〈/mrow〉〈/msup〉〈/math〉, there exists a pseudocompact Abelian group 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.gif" overflow="scroll"〉〈mi〉G〈/mi〉〈mo〉∈〈/mo〉〈mi〉C〈/mi〉〈mi〉S〈/mi〉〈mi〉S〈/mi〉〈/math〉 such that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.gif" overflow="scroll"〉〈mi〉G〈/mi〉〈mo〉×〈/mo〉〈mi〉G〈/mi〉〈mo〉∉〈/mo〉〈mi〉C〈/mi〉〈mi〉S〈/mi〉〈mi〉S〈/mi〉〈/math〉; (3) under the assumption of MA〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.gif" overflow="scroll"〉〈mspace width="0.2em"〉〈/mspace〉〈mi mathvariant="normal"〉&〈/mi〉〈mspace width="0.2em"〉〈/mspace〉〈mo〉¬〈/mo〉〈mspace width="0.2em"〉〈/mspace〉〈/math〉CH, there exist countably compact Abelian groups 〈em〉G〈/em〉 and 〈em〉H〈/em〉 in 〈em〉CSS〈/em〉 such that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.gif" overflow="scroll"〉〈mi〉G〈/mi〉〈mo〉×〈/mo〉〈mi〉H〈/mi〉〈mo〉∉〈/mo〉〈mi〉C〈/mi〉〈mi〉S〈/mi〉〈mi〉S〈/mi〉〈/math〉. Our example in (2) improves upon an example presented in the recent article [11] by A. Leiderman and the third listed author, while the example in (3) answers a question raised in that article.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 28 February 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications〈/p〉 〈p〉Author(s): James J. Madden, Martin Mugochi〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉A paratoplogical group is a group equipped with a topology in which multiplication, but not necessarily inversion, is continuous. An example is the Sorgenfry line under addition. In the present paper, we propose a reasonable point-free version of this notion – a paralocailc group. We provide several examples and non-examples, and we list several unsolved problems. Among our examples, we show that addition does not induce the structure of a paralocailc group on the frame of opens of the Sorgenfrey line. This is related to the non-spatiality of the coproduct the Sorgenfry topology with itself.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 28 February 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications〈/p〉 〈p〉Author(s): Wolfgang Herfort, Karl H. Hofmann, Francesco G. Russo〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉 〈p〉A locally compact abelian group is called 〈em〉periodic〈/em〉 if it is totally disconnected and is a directed union of its compact subgroups. Various aspects of abelian periodic groups are considered such as〈/p〉 〈dl〉 〈dt〉–〈/dt〉 〈dd〉〈p〉decomposing them into local products of their Sylow 〈em〉p〈/em〉- subgroups,〈/p〉〈/dd〉 〈dt〉–〈/dt〉 〈dd〉〈p〉providing new descriptions of periodic abelian torsion groups, and of〈/p〉〈/dd〉 〈dt〉–〈/dt〉 〈dd〉〈p〉periodic abelian divisible groups and their torsion-free and their torsion components,〈/p〉〈/dd〉 〈dt〉–〈/dt〉 〈dd〉〈p〉reviewing splitting theorems, notably for finite rank pure subgroups of (almost) finite exponent 〈em〉p〈/em〉-groups,〈/p〉〈/dd〉 〈dt〉–〈/dt〉 〈dd〉〈p〉providing a definition of a general 〈em〉p〈/em〉-rank for 〈em〉all〈/em〉 locally compact abelian 〈em〉p〈/em〉-groups.〈/p〉〈/dd〉 〈/dl〉 〈/div〉

Description: 〈p〉Publication date: Available online 28 February 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications〈/p〉 〈p〉Author(s): M. Hušek, J. Rosický〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Investigating dual local presentability of some topological and uniform classes, a new procedure is developed for factorization of maps defined on subspaces of products and a new characterization of local presentability is produced. The factorization is related to large cardinals and deals, mainly, with realcompact spaces. Instead of factorization of maps on colimits, local presentability is characterized by means of factorization on products.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 1 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications〈/p〉 〈p〉Author(s): Alan Dow〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉W. W. Comfort, in collaboration with A. Kato and S. Shelah, explored the problem of determining which spaces 〈em〉Y〈/em〉 satisfy that for every 2-coloring of 〈em〉βω〈/em〉 there will be a monochromatic copy of 〈em〉Y〈/em〉. We continue the exploration and answer some questions raised in their paper.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 28 February 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications〈/p〉 〈p〉Author(s): William Wistar Comfort, Dikran Dikranjan〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉 〈p〉Given a compact abelian group 〈em〉G〈/em〉, we study the poset 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi mathvariant="script"〉G〈/mi〉〈/mrow〉〈mrow〉〈mi〉δ〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉G〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si172.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉G〈/mi〉〈/mrow〉〈mrow〉〈mi〉δ〈/mi〉〈/mrow〉〈/msub〉〈/math〉-dense subgroups of 〈em〉G〈/em〉 and the subgroup 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.gif" overflow="scroll"〉〈mi〉δ〈/mi〉〈mtext〉-〈/mtext〉〈mrow〉〈mi mathvariant="bold"〉den〈/mi〉〈/mrow〉〈mo stretchy="false"〉(〈/mo〉〈mi〉G〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo〉:〈/mo〉〈mo〉=〈/mo〉〈mo〉⋂〈/mo〉〈msub〉〈mrow〉〈mi mathvariant="script"〉G〈/mi〉〈/mrow〉〈mrow〉〈mi〉δ〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉G〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉, named 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si172.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉G〈/mi〉〈/mrow〉〈mrow〉〈mi mathvariant="italic"〉δ〈/mi〉〈/mrow〉〈/msub〉〈/math〉〈em〉-density nucleus〈/em〉 of 〈em〉G〈/em〉. It turns out that for every compact abelian group 〈em〉G〈/em〉:〈/p〉 〈p〉(i) the subgroup 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si329.gif" overflow="scroll"〉〈mi〉δ〈/mi〉〈mtext〉-〈/mtext〉〈mrow〉〈mi mathvariant="bold"〉den〈/mi〉〈/mrow〉〈mo stretchy="false"〉(〈/mo〉〈mi〉G〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 is compact metrizable and coincides with the intersection of just two members 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉D〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈mo〉,〈/mo〉〈msub〉〈mrow〉〈mi〉D〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mo〉∈〈/mo〉〈msub〉〈mrow〉〈mi mathvariant="script"〉G〈/mi〉〈/mrow〉〈mrow〉〈mi〉δ〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉G〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉;〈/p〉 〈p〉(ii) if 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si7.gif" overflow="scroll"〉〈mi〉δ〈/mi〉〈mtext〉-〈/mtext〉〈mrow〉〈mi mathvariant="bold"〉den〈/mi〉〈/mrow〉〈mo stretchy="false"〉(〈/mo〉〈mi〉G〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo〉=〈/mo〉〈mo stretchy="false"〉{〈/mo〉〈mn〉0〈/mn〉〈mo stretchy="false"〉}〈/mo〉〈/math〉, then there exists some independent family 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si8.gif" overflow="scroll"〉〈mi mathvariant="script"〉F〈/mi〉〈/math〉 in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi mathvariant="script"〉G〈/mi〉〈/mrow〉〈mrow〉〈mi〉δ〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉G〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 (i.e., with 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.gif" overflow="scroll"〉〈mo stretchy="false"〉〈〈/mo〉〈mo〉⋃〈/mo〉〈mi mathvariant="script"〉F〈/mi〉〈mo stretchy="false"〉〉〈/mo〉〈mo〉=〈/mo〉〈msub〉〈mrow〉〈mo〉⨁〈/mo〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈mo〉∈〈/mo〉〈mi〉I〈/mi〉〈/mrow〉〈/msub〉〈msub〉〈mrow〉〈mi〉D〈/mi〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈/mrow〉〈/msub〉〈/math〉) such that:〈/p〉 〈p〉 (ii〈sub〉1〈/sub〉) 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.gif" overflow="scroll"〉〈mo stretchy="false"〉|〈/mo〉〈mi mathvariant="script"〉F〈/mi〉〈mo stretchy="false"〉|〈/mo〉〈mo〉=〈/mo〉〈mi〉r〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉G〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉, the free rank of 〈em〉G〈/em〉, if 〈em〉mG〈/em〉 is not metrizable for any positive 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si12.gif" overflow="scroll"〉〈mi〉m〈/mi〉〈mo〉∈〈/mo〉〈mi mathvariant="double-struck"〉N〈/mi〉〈/math〉; otherwise,〈/p〉 〈p〉 (ii〈sub〉2〈/sub〉) when 〈em〉G〈/em〉 is (necessarily) bounded torsion, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.gif" overflow="scroll"〉〈mo stretchy="false"〉|〈/mo〉〈mi mathvariant="script"〉F〈/mi〉〈mo stretchy="false"〉|〈/mo〉〈/math〉 coincides with the least leading Ulm-Kaplansky invariant of 〈em〉G〈/em〉.〈/p〉 〈p〉(iii) in option (ii〈sub〉1〈/sub〉) the members of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si8.gif" overflow="scroll"〉〈mi mathvariant="script"〉F〈/mi〉〈/math〉 can be chosen to be free subgroups of 〈em〉G〈/em〉 precisely when 〈em〉G〈/em〉 admits a free dense subgroup (equivalently, when 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si14.gif" overflow="scroll"〉〈mi〉r〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉G〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo〉≥〈/mo〉〈mi〉d〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉G〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉); 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si8.gif" overflow="scroll"〉〈mi mathvariant="script"〉F〈/mi〉〈/math〉 can have (the maximum possible) size 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.gif" overflow="scroll"〉〈mo stretchy="false"〉|〈/mo〉〈mi〉G〈/mi〉〈mo stretchy="false"〉|〈/mo〉〈/math〉 if and only if 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.gif" overflow="scroll"〉〈mi〉r〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉G〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo〉=〈/mo〉〈mo stretchy="false"〉|〈/mo〉〈mi〉G〈/mi〉〈mo stretchy="false"〉|〈/mo〉〈/math〉;〈/p〉 〈p〉(iv) in option (ii〈sub〉2〈/sub〉) the family 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si8.gif" overflow="scroll"〉〈mi mathvariant="script"〉F〈/mi〉〈/math〉 can have size 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.gif" overflow="scroll"〉〈mo stretchy="false"〉|〈/mo〉〈mi〉G〈/mi〉〈mo stretchy="false"〉|〈/mo〉〈/math〉 if and only if all leading Ulm-Kaplansky invariants of 〈em〉G〈/em〉 coincide with 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.gif" overflow="scroll"〉〈mo stretchy="false"〉|〈/mo〉〈mi〉G〈/mi〉〈mo stretchy="false"〉|〈/mo〉〈/math〉.〈/p〉 〈p〉Resuming, the compact abelian groups 〈em〉G〈/em〉 with trivial 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si172.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉G〈/mi〉〈/mrow〉〈mrow〉〈mi〉δ〈/mi〉〈/mrow〉〈/msub〉〈/math〉-density nucleus 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si329.gif" overflow="scroll"〉〈mi〉δ〈/mi〉〈mtext〉-〈/mtext〉〈mrow〉〈mi mathvariant="bold"〉den〈/mi〉〈/mrow〉〈mo stretchy="false"〉(〈/mo〉〈mi〉G〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 are, roughly speaking, “maximally fragmented” into 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si172.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉G〈/mi〉〈/mrow〉〈mrow〉〈mi〉δ〈/mi〉〈/mrow〉〈/msub〉〈/math〉-dense subgroups. More precisely, the existence of a pair of subgroups 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉D〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈mo〉,〈/mo〉〈msub〉〈mrow〉〈mi〉D〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mo〉∈〈/mo〉〈msub〉〈mrow〉〈mi mathvariant="script"〉G〈/mi〉〈/mrow〉〈mrow〉〈mi〉δ〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉G〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 with 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si17.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉D〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msub〉〈mo〉∩〈/mo〉〈msub〉〈mrow〉〈mi〉D〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mo〉=〈/mo〉〈mo stretchy="false"〉{〈/mo〉〈mn〉0〈/mn〉〈mo stretchy="false"〉}〈/mo〉〈/math〉 yields the existence of a large independent family 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si18.gif" overflow="scroll"〉〈mi mathvariant="script"〉F〈/mi〉〈mo〉=〈/mo〉〈mo stretchy="false"〉{〈/mo〉〈msub〉〈mrow〉〈mi〉D〈/mi〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈/mrow〉〈/msub〉〈mo〉∈〈/mo〉〈mi〉I〈/mi〉〈mo stretchy="false"〉}〈/mo〉〈mo〉⊆〈/mo〉〈msub〉〈mrow〉〈mi mathvariant="script"〉G〈/mi〉〈/mrow〉〈mrow〉〈mi〉δ〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉G〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 with 〈em〉I〈/em〉 is as big as possible (i.e., as big as the algebraic structure of 〈em〉G〈/em〉 permits in terms of some immediate necessary restraints involving relevant algebraic cardinal invariants of the group, in the spirit of those in (ii)–(iv)).〈/p〉 〈/div〉

Description: 〈p〉Publication date: Available online 28 February 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications〈/p〉 〈p〉Author(s): Dieter Remus, Mihail Ursul〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, results of Tsarelunga resp. Comfort, Szambien and the first-listed author are improved. Throughout this abstract, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mo stretchy="false"〉(〈/mo〉〈mi〉R〈/mi〉〈mo〉,〈/mo〉〈mi mathvariant="script"〉T〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 denotes a nonmetrizable compact ring. First a main tool is shown: If 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mo stretchy="false"〉(〈/mo〉〈mi〉R〈/mi〉〈mo〉,〈/mo〉〈mi mathvariant="script"〉T〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 is topologically nilpotent, then 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si165.gif" overflow="scroll"〉〈mi〉w〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉R〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo〉=〈/mo〉〈mi〉w〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉R〈/mi〉〈mo stretchy="false"〉/〈/mo〉〈mover accent="true"〉〈mrow〉〈msup〉〈mrow〉〈mi〉R〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈/mrow〉〈mo〉‾〈/mo〉〈/mover〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 holds. By using tensor products of unitary modules it is proved that every nonmetrizable compact ring with an identity has a proper pseudocompact refinement. 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mo stretchy="false"〉(〈/mo〉〈mi〉R〈/mi〉〈mo〉,〈/mo〉〈mi mathvariant="script"〉T〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 admits exactly 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈mrow〉〈msup〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈mrow〉〈mo stretchy="false"〉|〈/mo〉〈mi〉R〈/mi〉〈mo stretchy="false"〉|〈/mo〉〈/mrow〉〈/msup〉〈/mrow〉〈/msup〉〈/math〉-many pseudocompact ring topologies on 〈em〉R〈/em〉 finer than 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.gif" overflow="scroll"〉〈mi mathvariant="script"〉T〈/mi〉〈/math〉 in the following cases: 〈em〉R〈/em〉 is a commutative local ring; 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mo stretchy="false"〉(〈/mo〉〈mi〉R〈/mi〉〈mo〉,〈/mo〉〈mi mathvariant="script"〉T〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 is topologically nilpotent; 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mo stretchy="false"〉(〈/mo〉〈mi〉R〈/mi〉〈mo〉,〈/mo〉〈mi mathvariant="script"〉T〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 is commutative such that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.gif" overflow="scroll"〉〈mi〉w〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉R〈/mi〉〈mo〉,〈/mo〉〈mi mathvariant="script"〉T〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 is a regular cardinal number.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 28 February 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications〈/p〉 〈p〉Author(s): Osvaldo Guzmán, Michael Hrušák〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We study a class of ideals introduced by Pospíšil. We answer a question of the second author by proving that there is an 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉F〈/mi〉〈/mrow〉〈mrow〉〈mi〉σ〈/mi〉〈mi〉δ〈/mi〉〈mi〉σ〈/mi〉〈/mrow〉〈/msub〉〈/math〉 ideal 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.gif" overflow="scroll"〉〈mi mathvariant="script"〉I〈/mi〉〈/math〉 such that every filter of character less than 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si240.gif" overflow="scroll"〉〈mi mathvariant="fraktur"〉c〈/mi〉〈/math〉 can be extended to an 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.gif" overflow="scroll"〉〈mi mathvariant="script"〉I〈/mi〉〈/math〉-ultrafilter. We also prove that this statement is consistently false for 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si277.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉F〈/mi〉〈/mrow〉〈mrow〉〈mi〉σ〈/mi〉〈mi〉δ〈/mi〉〈/mrow〉〈/msub〉〈/math〉-ideals.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 28 February 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications〈/p〉 〈p〉Author(s): Neil Hindman, Dona Strauss〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Let 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mo stretchy="false"〉(〈/mo〉〈mi〉S〈/mi〉〈mo〉,〈/mo〉〈mo〉+〈/mo〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 be an infinite commutative semigroup with identity 0. Let 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈mi〉u〈/mi〉〈mo〉,〈/mo〉〈mi〉v〈/mi〉〈mo〉∈〈/mo〉〈mi mathvariant="double-struck"〉N〈/mi〉〈/math〉 and let 〈em〉A〈/em〉 be a 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.gif" overflow="scroll"〉〈mi〉u〈/mi〉〈mo〉×〈/mo〉〈mi〉v〈/mi〉〈/math〉 matrix with nonnegative integer entries. If 〈em〉S〈/em〉 is cancellative, let the entries of 〈em〉A〈/em〉 come from 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si295.gif" overflow="scroll"〉〈mi mathvariant="double-struck"〉Z〈/mi〉〈/math〉. Then 〈em〉A〈/em〉 is 〈em〉image partition regular over S〈/em〉 (〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si449.gif" overflow="scroll"〉〈mi〉I〈/mi〉〈mi〉P〈/mi〉〈mi〉R〈/mi〉〈mo stretchy="false"〉/〈/mo〉〈mi〉S〈/mi〉〈/math〉) iff whenever 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si450.gif" overflow="scroll"〉〈mi〉S〈/mi〉〈mo〉∖〈/mo〉〈mo stretchy="false"〉{〈/mo〉〈mn〉0〈/mn〉〈mo stretchy="false"〉}〈/mo〉〈/math〉 is finitely colored, there exists 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.gif" overflow="scroll"〉〈mover accent="true"〉〈mrow〉〈mi〉x〈/mi〉〈/mrow〉〈mrow〉〈mo stretchy="false"〉→〈/mo〉〈/mrow〉〈/mover〉〈mo〉∈〈/mo〉〈msup〉〈mrow〉〈mo stretchy="false"〉(〈/mo〉〈mi〉S〈/mi〉〈mo〉∖〈/mo〉〈mo stretchy="false"〉{〈/mo〉〈mn〉0〈/mn〉〈mo stretchy="false"〉}〈/mo〉〈mo stretchy="false"〉)〈/mo〉〈/mrow〉〈mrow〉〈mi〉v〈/mi〉〈/mrow〉〈/msup〉〈/math〉 such that the entries of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si406.gif" overflow="scroll"〉〈mi〉A〈/mi〉〈mover accent="true"〉〈mrow〉〈mi〉x〈/mi〉〈/mrow〉〈mrow〉〈mo stretchy="false"〉→〈/mo〉〈/mrow〉〈/mover〉〈/math〉 are monochromatic. The matrix 〈em〉A〈/em〉 is 〈em〉centrally image partition regular over S〈/em〉 (〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si317.gif" overflow="scroll"〉〈mi〉C〈/mi〉〈mi〉I〈/mi〉〈mi〉P〈/mi〉〈mi〉R〈/mi〉〈mo stretchy="false"〉/〈/mo〉〈mi〉S〈/mi〉〈/math〉) iff whenever 〈em〉C〈/em〉 is a central subset of 〈em〉S〈/em〉, there exists 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.gif" overflow="scroll"〉〈mover accent="true"〉〈mrow〉〈mi〉x〈/mi〉〈/mrow〉〈mrow〉〈mo stretchy="false"〉→〈/mo〉〈/mrow〉〈/mover〉〈mo〉∈〈/mo〉〈msup〉〈mrow〉〈mo stretchy="false"〉(〈/mo〉〈mi〉S〈/mi〉〈mo〉∖〈/mo〉〈mo stretchy="false"〉{〈/mo〉〈mn〉0〈/mn〉〈mo stretchy="false"〉}〈/mo〉〈mo stretchy="false"〉)〈/mo〉〈/mrow〉〈mrow〉〈mi〉v〈/mi〉〈/mrow〉〈/msup〉〈/math〉 such that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.gif" overflow="scroll"〉〈mi〉A〈/mi〉〈mover accent="true"〉〈mrow〉〈mi〉x〈/mi〉〈/mrow〉〈mrow〉〈mo stretchy="false"〉→〈/mo〉〈/mrow〉〈/mover〉〈mo〉∈〈/mo〉〈msup〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈/msup〉〈/math〉. These notions have been extensively studied for subsemigroups of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si12.gif" overflow="scroll"〉〈mo stretchy="false"〉(〈/mo〉〈mi mathvariant="double-struck"〉R〈/mi〉〈mo〉,〈/mo〉〈mo〉+〈/mo〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 or 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si13.gif" overflow="scroll"〉〈mo stretchy="false"〉(〈/mo〉〈mi mathvariant="double-struck"〉R〈/mi〉〈mo〉,〈/mo〉〈mo〉⋅〈/mo〉〈mo stretchy="false"〉)〈/mo〉〈/math〉. We obtain some necessary and some sufficient conditions for 〈em〉A〈/em〉 to be 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si449.gif" overflow="scroll"〉〈mi〉I〈/mi〉〈mi〉P〈/mi〉〈mi〉R〈/mi〉〈mo stretchy="false"〉/〈/mo〉〈mi〉S〈/mi〉〈/math〉 or 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si317.gif" overflow="scroll"〉〈mi〉C〈/mi〉〈mi〉I〈/mi〉〈mi〉P〈/mi〉〈mi〉R〈/mi〉〈mo stretchy="false"〉/〈/mo〉〈mi〉S〈/mi〉〈/math〉. For example, if 〈em〉G〈/em〉 is an infinite divisible group, then 〈em〉A〈/em〉 is 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si14.gif" overflow="scroll"〉〈mi〉C〈/mi〉〈mi〉I〈/mi〉〈mi〉P〈/mi〉〈mi〉R〈/mi〉〈mo stretchy="false"〉/〈/mo〉〈mi〉G〈/mi〉〈/math〉 iff 〈em〉A〈/em〉 is 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.gif" overflow="scroll"〉〈mi〉I〈/mi〉〈mi〉P〈/mi〉〈mi〉R〈/mi〉〈mo stretchy="false"〉/〈/mo〉〈mi mathvariant="double-struck"〉Z〈/mi〉〈/math〉. If for all 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.gif" overflow="scroll"〉〈mi〉c〈/mi〉〈mo〉∈〈/mo〉〈mi mathvariant="double-struck"〉N〈/mi〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si17.gif" overflow="scroll"〉〈mi〉c〈/mi〉〈mi〉S〈/mi〉〈mo〉≠〈/mo〉〈mo stretchy="false"〉{〈/mo〉〈mn〉0〈/mn〉〈mo stretchy="false"〉}〈/mo〉〈/math〉 and 〈em〉A〈/em〉 is 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si18.gif" overflow="scroll"〉〈mi〉I〈/mi〉〈mi〉P〈/mi〉〈mi〉R〈/mi〉〈mo stretchy="false"〉/〈/mo〉〈mi mathvariant="double-struck"〉N〈/mi〉〈/math〉, then 〈em〉A〈/em〉 is 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si449.gif" overflow="scroll"〉〈mi〉I〈/mi〉〈mi〉P〈/mi〉〈mi〉R〈/mi〉〈mo stretchy="false"〉/〈/mo〉〈mi〉S〈/mi〉〈/math〉. If 〈em〉S〈/em〉 is cancellative, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.gif" overflow="scroll"〉〈mi〉c〈/mi〉〈mo〉∈〈/mo〉〈mi mathvariant="double-struck"〉N〈/mi〉〈/math〉, and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si19.gif" overflow="scroll"〉〈mi〉c〈/mi〉〈mi〉S〈/mi〉〈mo〉=〈/mo〉〈mo stretchy="false"〉{〈/mo〉〈mn〉0〈/mn〉〈mo stretchy="false"〉}〈/mo〉〈/math〉, we obtain a simple sufficient condition for 〈em〉A〈/em〉 to be 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si449.gif" overflow="scroll"〉〈mi〉I〈/mi〉〈mi〉P〈/mi〉〈mi〉R〈/mi〉〈mo stretchy="false"〉/〈/mo〉〈mi〉S〈/mi〉〈/math〉. It is well-known that 〈em〉A〈/em〉 is 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si449.gif" overflow="scroll"〉〈mi〉I〈/mi〉〈mi〉P〈/mi〉〈mi〉R〈/mi〉〈mo stretchy="false"〉/〈/mo〉〈mi〉S〈/mi〉〈/math〉 if 〈em〉A〈/em〉 is a first entries matrix with the property that 〈em〉cS〈/em〉 is a central〈sup〉⁎〈/sup〉 subset of 〈em〉S〈/em〉 for every first entry 〈em〉c〈/em〉 of 〈em〉A〈/em〉. We extend this theorem to first entries matrices whose first entries may not satisfy this condition. We discuss whether, if 〈em〉S〈/em〉 is finitely colored, there exists 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.gif" overflow="scroll"〉〈mover accent="true"〉〈mrow〉〈mi〉x〈/mi〉〈/mrow〉〈mrow〉〈mo stretchy="false"〉→〈/mo〉〈/mrow〉〈/mover〉〈mo〉∈〈/mo〉〈msup〉〈mrow〉〈mo stretchy="false"〉(〈/mo〉〈mi〉S〈/mi〉〈mo〉∖〈/mo〉〈mo stretchy="false"〉{〈/mo〉〈mn〉0〈/mn〉〈mo stretchy="false"〉}〈/mo〉〈mo stretchy="false"〉)〈/mo〉〈/mrow〉〈mrow〉〈mi〉v〈/mi〉〈/mrow〉〈/msup〉〈/math〉, with distinct entries, for which the entries of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si406.gif" overflow="scroll"〉〈mi〉A〈/mi〉〈mover accent="true"〉〈mrow〉〈mi〉x〈/mi〉〈/mrow〉〈mrow〉〈mo stretchy="false"〉→〈/mo〉〈/mrow〉〈/mover〉〈/math〉 are monochromatic and distinct. Along the way, we obtain several new results about the algebra of 〈em〉βS〈/em〉, the Stone-Čech compactification of the discrete semigroup 〈em〉S〈/em〉.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 28 February 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications〈/p〉 〈p〉Author(s): Philip Scowcroft〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉After establishing a completeness theorem for continuous logic, Ben Yaacov and Pedersen conclude that if 〈em〉T〈/em〉 is a complete recursive 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mi mathvariant="script"〉L〈/mi〉〈/math〉-theory in continuous logic, and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈mi〉v〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉φ〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 is the truth value of the 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mi mathvariant="script"〉L〈/mi〉〈/math〉-sentence 〈em〉φ〈/em〉 in models of 〈em〉T〈/em〉, then 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈mi〉v〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉φ〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 is a recursive real uniformly recursive in 〈em〉φ〈/em〉. Some of the examples to which the latter result applies are theories of the following structures: atomless probability structures, the Urysohn space of diameter 1, Hilbert space, the lattice-ordered group or ring of real-valued continuous functions on the Cantor set, and the complex 〈sup〉⁎〈/sup〉-algebra of continuous functions on the Cantor set. This paper will explain why these examples obey much stronger results, yielding (for example) decision procedures for the conditions true in these structures.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 27 February 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications〈/p〉 〈p〉Author(s): George Baloglou, Johannes Vermeer〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The authors, University of Kansas post-doctoral instructors at the time this note was written, exhibit an upper bound for the Lindelöf degree of a product space.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 26 April 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications〈/p〉 〈p〉Author(s): Zipei Nie〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, we prove that the fundamental group of the manifold obtained by Dehn surgery along a 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mo stretchy="false"〉(〈/mo〉〈mo〉−〈/mo〉〈mn〉2〈/mn〉〈mo〉,〈/mo〉〈mn〉3〈/mn〉〈mo〉,〈/mo〉〈mn〉2〈/mn〉〈mi〉s〈/mi〉〈mo〉+〈/mo〉〈mn〉1〈/mn〉〈mo stretchy="false"〉)〈/mo〉〈/math〉-pretzel knot (〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si14.gif" overflow="scroll"〉〈mi〉s〈/mi〉〈mo〉≥〈/mo〉〈mn〉3〈/mn〉〈/math〉) with slope 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.gif" overflow="scroll"〉〈mfrac〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈mrow〉〈mi〉q〈/mi〉〈/mrow〉〈/mfrac〉〈/math〉 is not left orderable if 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si22.gif" overflow="scroll"〉〈mfrac〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈mrow〉〈mi〉q〈/mi〉〈/mrow〉〈/mfrac〉〈mo〉≥〈/mo〉〈mn〉2〈/mn〉〈mi〉s〈/mi〉〈mo〉+〈/mo〉〈mn〉3〈/mn〉〈/math〉, and that it is left orderable if 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.gif" overflow="scroll"〉〈mfrac〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈mrow〉〈mi〉q〈/mi〉〈/mrow〉〈/mfrac〉〈/math〉 is in a neighborhood of zero depending on 〈em〉s〈/em〉.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 24 April 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications〈/p〉 〈p〉Author(s): Mirosław Ślosarski〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉This article provides a general method of the construction of relative retracts (in particular, multiretracts), approximative relative retracts and multi-retracts in the sense of Suszycki.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 June 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 260〈/p〉 〈p〉Author(s): F. Casarrubias-Segura, R. Rojas-Hernández〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We provide an example to show that the monotone Sokolov property is not necessarily preserved under compact continuous images. Furthermore, we prove that if 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mi〉X〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mo〉∖〈/mo〉〈msub〉〈mrow〉〈mi mathvariant="normal"〉Δ〈/mi〉〈/mrow〉〈mrow〉〈mi〉X〈/mi〉〈/mrow〉〈/msub〉〈/math〉 is either monotonically Sokolov or monotonically retractable, then 〈em〉X〈/em〉 must be cosmic; and that if 〈em〉X〈/em〉 is either hereditarily monotonically Sokolov or hereditarily monotonically retractable, then 〈em〉X〈/em〉 has a countable network. Moreover, we characterize the Lindelöf Σ-property in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msub〉〈/math〉-spaces on Alexandrov doubles, and show that the 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.gif" overflow="scroll"〉〈mi〉L〈/mi〉〈mi mathvariant="normal"〉Σ〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉L〈/mi〉〈mi mathvariant="normal"〉Σ〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mo〉⩽〈/mo〉〈mi〉ω〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo stretchy="false"〉)〈/mo〉〈/math〉-property is equivalent to the 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si12.gif" overflow="scroll"〉〈mi〉L〈/mi〉〈mi mathvariant="normal"〉Σ〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mo〉⩽〈/mo〉〈mi〉ω〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉-property for the class of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msub〉〈/math〉-spaces and for the class of Gul'ko spaces. These results solve some problems published by Kalenda [7], Tkachuk [23], García-Ferreira and Rojas-Hernández [5], and Molina-Lara and Okunev [11].〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 June 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 260〈/p〉 〈p〉Author(s): Taras Banakh, Serhii Bardyla〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We find (completeness type) conditions on topological semilattices 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mi〉X〈/mi〉〈mo〉,〈/mo〉〈mi〉Y〈/mi〉〈/math〉 guaranteeing that each continuous homomorphism 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈mi〉h〈/mi〉〈mo〉:〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉→〈/mo〉〈mi〉Y〈/mi〉〈/math〉 has closed image 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.gif" overflow="scroll"〉〈mi〉h〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 in 〈em〉Y〈/em〉.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 May 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 258〈/p〉 〈p〉Author(s): Olga Frolkina〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Let 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈mi mathvariant="script"〉C〈/mi〉〈/math〉 be the Cantor set. For each 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si18.gif" overflow="scroll"〉〈mi〉n〈/mi〉〈mo〉⩾〈/mo〉〈mn〉3〈/mn〉〈/math〉 we construct an embedding 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si32.gif" overflow="scroll"〉〈mi〉A〈/mi〉〈mo〉:〈/mo〉〈mi mathvariant="script"〉C〈/mi〉〈mo〉×〈/mo〉〈mi mathvariant="script"〉C〈/mi〉〈mo stretchy="false"〉→〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msup〉〈/math〉 such that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.gif" overflow="scroll"〉〈mi〉A〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi mathvariant="script"〉C〈/mi〉〈mo〉×〈/mo〉〈mo stretchy="false"〉{〈/mo〉〈mi〉s〈/mi〉〈mo stretchy="false"〉}〈/mo〉〈mo stretchy="false"〉)〈/mo〉〈/math〉, for 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si22.gif" overflow="scroll"〉〈mi〉s〈/mi〉〈mo〉∈〈/mo〉〈mi mathvariant="script"〉C〈/mi〉〈/math〉, are pairwise ambiently incomparable everywhere wild Cantor sets (generalized Antoine's necklaces). This serves as a base for another new result proved in this paper: for each 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si18.gif" overflow="scroll"〉〈mi〉n〈/mi〉〈mo〉⩾〈/mo〉〈mn〉3〈/mn〉〈/math〉 and any non-empty perfect compact set 〈em〉X〈/em〉 which is embeddable in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si125.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈mo〉−〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈/msup〉〈/math〉, we describe an embedding 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si20.gif" overflow="scroll"〉〈mi mathvariant="double-struck"〉A〈/mi〉〈mo〉:〈/mo〉〈mi〉X〈/mi〉〈mo〉×〈/mo〉〈mi mathvariant="script"〉C〈/mi〉〈mo stretchy="false"〉→〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msup〉〈/math〉 such that each 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si113.gif" overflow="scroll"〉〈mi mathvariant="double-struck"〉A〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo〉×〈/mo〉〈mo stretchy="false"〉{〈/mo〉〈mi〉s〈/mi〉〈mo stretchy="false"〉}〈/mo〉〈mo stretchy="false"〉)〈/mo〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si22.gif" overflow="scroll"〉〈mi〉s〈/mi〉〈mo〉∈〈/mo〉〈mi mathvariant="script"〉C〈/mi〉〈/math〉, contains the corresponding 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.gif" overflow="scroll"〉〈mi〉A〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi mathvariant="script"〉C〈/mi〉〈mo〉×〈/mo〉〈mo stretchy="false"〉{〈/mo〉〈mi〉s〈/mi〉〈mo stretchy="false"〉}〈/mo〉〈mo stretchy="false"〉)〈/mo〉〈/math〉, and is “nice” on the complement 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.gif" overflow="scroll"〉〈mi mathvariant="double-struck"〉A〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo〉×〈/mo〉〈mo stretchy="false"〉{〈/mo〉〈mi〉s〈/mi〉〈mo stretchy="false"〉}〈/mo〉〈mo stretchy="false"〉)〈/mo〉〈mo〉−〈/mo〉〈mi〉A〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi mathvariant="script"〉C〈/mi〉〈mo〉×〈/mo〉〈mo stretchy="false"〉{〈/mo〉〈mi〉s〈/mi〉〈mo stretchy="false"〉}〈/mo〉〈mo stretchy="false"〉)〈/mo〉〈/math〉; in particular, the images 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si113.gif" overflow="scroll"〉〈mi mathvariant="double-struck"〉A〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo〉×〈/mo〉〈mo stretchy="false"〉{〈/mo〉〈mi〉s〈/mi〉〈mo stretchy="false"〉}〈/mo〉〈mo stretchy="false"〉)〈/mo〉〈/math〉, for 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si22.gif" overflow="scroll"〉〈mi〉s〈/mi〉〈mo〉∈〈/mo〉〈mi mathvariant="script"〉C〈/mi〉〈/math〉, are ambiently incomparable pairwise disjoint copies of 〈em〉X〈/em〉. This generalizes and strengthens theorems of J.R. Stallings (1960), R.B. Sher (1968), and B.L. Brechner–J.C. Mayer (1988).〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 May 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 258〈/p〉 〈p〉Author(s): Kun Du〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Suppose 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mi〉M〈/mi〉〈mo〉=〈/mo〉〈mi〉V〈/mi〉〈msub〉〈mrow〉〈mo〉∪〈/mo〉〈/mrow〉〈mrow〉〈mi〉S〈/mi〉〈/mrow〉〈/msub〉〈mi〉W〈/mi〉〈/math〉 is a Heegaard splitting of 〈em〉M〈/em〉 where 〈em〉V〈/em〉 contains the only one essential disk 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si105.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉D〈/mi〉〈/mrow〉〈mrow〉〈mi〉V〈/mi〉〈/mrow〉〈/msub〉〈/math〉 in 〈em〉V〈/em〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si106.gif" overflow="scroll"〉〈mi〉d〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉S〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo〉=〈/mo〉〈mi〉n〈/mi〉〈/math〉. Then, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si105.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉D〈/mi〉〈/mrow〉〈mrow〉〈mi〉V〈/mi〉〈/mrow〉〈/msub〉〈/math〉 is separating and cuts 〈em〉V〈/em〉 into two trivial compression bodies 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉V〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉V〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈/math〉. Suppose 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉F〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈/math〉 is a component of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si7.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mo〉∂〈/mo〉〈/mrow〉〈mrow〉〈mo〉−〈/mo〉〈/mrow〉〈/msub〉〈mi〉V〈/mi〉〈/math〉 such that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si8.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉F〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mo〉=〈/mo〉〈msub〉〈mrow〉〈mo〉∂〈/mo〉〈/mrow〉〈mrow〉〈mo〉−〈/mo〉〈/mrow〉〈/msub〉〈msub〉〈mrow〉〈mi〉V〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si87.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉S〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mo〉=〈/mo〉〈msub〉〈mrow〉〈mo〉∂〈/mo〉〈/mrow〉〈mrow〉〈mo〉+〈/mo〉〈/mrow〉〈/msub〉〈msub〉〈mrow〉〈mi〉V〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mo〉∩〈/mo〉〈mi〉S〈/mi〉〈/math〉. Then, we can define 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉ψ〈/mi〉〈/mrow〉〈mrow〉〈msub〉〈mrow〉〈mi〉F〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈/mrow〉〈/msub〉〈mo〉:〈/mo〉〈mi mathvariant="script"〉C〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉S〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo stretchy="false"〉→〈/mo〉〈mi mathvariant="script"〉C〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈msub〉〈mrow〉〈mi〉F〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉)〈/mo〉〈/math〉. Suppose 〈em〉X〈/em〉 is a full simplex of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.gif" overflow="scroll"〉〈mi mathvariant="script"〉C〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈msub〉〈mrow〉〈mi〉F〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉)〈/mo〉〈/math〉, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si12.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉H〈/mi〉〈/mrow〉〈mrow〉〈mi〉X〈/mi〉〈/mrow〉〈/msub〉〈/math〉 is a handlebody obtained by attaching 2-handles to 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉F〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈/math〉 along 〈em〉X〈/em〉 and capping off possible 2-spheres by 3-handles. We denote 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si101.gif" overflow="scroll"〉〈mi〉V〈/mi〉〈mo〉∪〈/mo〉〈msub〉〈mrow〉〈mi〉H〈/mi〉〈/mrow〉〈mrow〉〈mi〉X〈/mi〉〈/mrow〉〈/msub〉〈/math〉 by 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si14.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉V〈/mi〉〈/mrow〉〈mrow〉〈mi〉H〈/mi〉〈/mrow〉〈/msub〉〈/math〉. Then, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si15.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉M〈/mi〉〈/mrow〉〈mrow〉〈mi〉H〈/mi〉〈/mrow〉〈/msub〉〈mo〉=〈/mo〉〈msub〉〈mrow〉〈mi〉V〈/mi〉〈/mrow〉〈mrow〉〈mi〉H〈/mi〉〈/mrow〉〈/msub〉〈msub〉〈mrow〉〈mo〉∪〈/mo〉〈/mrow〉〈mrow〉〈msub〉〈mrow〉〈mi〉S〈/mi〉〈/mrow〉〈mrow〉〈mi〉H〈/mi〉〈/mrow〉〈/msub〉〈/mrow〉〈/msub〉〈mi〉W〈/mi〉〈/math〉 is a Heegaard splitting of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si108.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉M〈/mi〉〈/mrow〉〈mrow〉〈mi〉H〈/mi〉〈/mrow〉〈/msub〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si108.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉M〈/mi〉〈/mrow〉〈mrow〉〈mi〉H〈/mi〉〈/mrow〉〈/msub〉〈/math〉 is said to be a handlebody filling of 〈em〉M〈/em〉. In this paper, we suppose that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si109.gif" overflow="scroll"〉〈mn〉0〈/mn〉〈mo〉〈〈/mo〉〈mi〉d〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈msub〉〈mrow〉〈mi〉S〈/mi〉〈/mrow〉〈mrow〉〈mi〉H〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉)〈/mo〉〈mo〉〈〈/mo〉〈mi〉n〈/mi〉〈/math〉, then there is a constant 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si110.gif" overflow="scroll"〉〈mi mathvariant="script"〉M〈/mi〉〈/math〉 such that either if 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si111.gif" overflow="scroll"〉〈mn〉0〈/mn〉〈mo〉〈〈/mo〉〈mi〉k〈/mi〉〈mo〉≤〈/mo〉〈mfrac〉〈mrow〉〈mi mathvariant="script"〉M〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/mfrac〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si112.gif" overflow="scroll"〉〈mn〉2〈/mn〉〈mi〉k〈/mi〉〈mo〉−〈/mo〉〈mn〉2〈/mn〉〈mo〉≤〈/mo〉〈msub〉〈mrow〉〈mi〉d〈/mi〉〈/mrow〉〈mrow〉〈mi mathvariant="script"〉C〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈msub〉〈mrow〉〈mi〉F〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉)〈/mo〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi mathvariant="script"〉D〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈msub〉〈mrow〉〈mi〉H〈/mi〉〈/mrow〉〈mrow〉〈mi〉X〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉)〈/mo〉〈mo〉,〈/mo〉〈msub〉〈mrow〉〈mi〉ψ〈/mi〉〈/mrow〉〈mrow〉〈msub〉〈mrow〉〈mi〉F〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈msub〉〈mrow〉〈mi mathvariant="script"〉D〈/mi〉〈/mrow〉〈mrow〉〈mi〉W〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉)〈/mo〉〈mo stretchy="false"〉)〈/mo〉〈mo〉≤〈/mo〉〈mn〉2〈/mn〉〈mi〉k〈/mi〉〈mo〉−〈/mo〉〈mn〉1〈/mn〉〈/math〉, then 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si21.gif" overflow="scroll"〉〈mi〉d〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈msub〉〈mrow〉〈mi〉S〈/mi〉〈/mrow〉〈mrow〉〈mi〉H〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉)〈/mo〉〈mo〉=〈/mo〉〈mi〉k〈/mi〉〈/math〉 or 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si113.gif" overflow="scroll"〉〈mfrac〉〈mrow〉〈mi mathvariant="script"〉M〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/mfrac〉〈mo〉〈〈/mo〉〈mi〉d〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈msub〉〈mrow〉〈mi〉S〈/mi〉〈/mrow〉〈mrow〉〈mi〉H〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉)〈/mo〉〈mo〉〈〈/mo〉〈mi〉n〈/mi〉〈/math〉.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 May 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 258〈/p〉 〈p〉Author(s): Á. Tamariz-Mascarúa, M.G. Tkachenko〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Let 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mi〉X〈/mi〉〈mo〉=〈/mo〉〈msub〉〈mrow〉〈mo〉∏〈/mo〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈mo〉∈〈/mo〉〈mi〉I〈/mi〉〈/mrow〉〈/msub〉〈msub〉〈mrow〉〈mi〉X〈/mi〉〈/mrow〉〈mrow〉〈mi〉i〈/mi〉〈/mrow〉〈/msub〉〈/math〉 be a product of non-compact spaces. We show that if 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈mo stretchy="false"〉|〈/mo〉〈mi〉I〈/mi〉〈mo stretchy="false"〉|〈/mo〉〈mo〉〉〈/mo〉〈mi〉ω〈/mi〉〈/math〉, then the remainder 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.gif" overflow="scroll"〉〈mi〉Y〈/mi〉〈mo〉=〈/mo〉〈mi〉b〈/mi〉〈mi〉X〈/mi〉〈mo〉∖〈/mo〉〈mi〉X〈/mi〉〈/math〉 is pseudocompact, for any compactification 〈em〉bX〈/em〉 of 〈em〉X〈/em〉. In fact, this theorem follows from a more general result about spaces with an 〈em〉ω-directed lattice〈/em〉 of 〈em〉d〈/em〉-open mappings. Under the additional assumption that the space 〈em〉X〈/em〉 has countable cellularity, we prove that the remainder 〈em〉Y〈/em〉 is 〈em〉C〈/em〉-embedded in 〈em〉bX〈/em〉 and that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si120.gif" overflow="scroll"〉〈mi〉β〈/mi〉〈mi〉Y〈/mi〉〈mo〉=〈/mo〉〈mi〉b〈/mi〉〈mi〉X〈/mi〉〈/math〉. We apply these results to the remainders of topological groups and spaces of continuous functions with the pointwise convergence topology. For example, we prove that if 〈em〉X〈/em〉 is an uncountable space and 〈em〉G〈/em〉 is a non-compact topological group, then every remainder of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si296.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo〉,〈/mo〉〈mi〉G〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 is pseudocompact provided 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si296.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo〉,〈/mo〉〈mi〉G〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 is dense in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.gif" overflow="scroll"〉〈msup〉〈mrow〉〈mi〉G〈/mi〉〈/mrow〉〈mrow〉〈mi〉X〈/mi〉〈/mrow〉〈/msup〉〈/math〉.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 May 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 258〈/p〉 〈p〉Author(s): Faze Zhang, Yanqing Zou〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉 〈p〉Let 〈em〉S〈/em〉 be an orientable closed surface with genus at least two. From 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mi〉S〈/mi〉〈mo〉×〈/mo〉〈mi〉I〈/mi〉〈/math〉, for a given orientation-reversing homeomorphism 〈em〉f〈/em〉 from 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈mi〉S〈/mi〉〈mo〉×〈/mo〉〈mo stretchy="false"〉{〈/mo〉〈mn〉1〈/mn〉〈mo stretchy="false"〉}〈/mo〉〈/math〉 to 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.gif" overflow="scroll"〉〈mi〉S〈/mi〉〈mo〉×〈/mo〉〈mo stretchy="false"〉{〈/mo〉〈mn〉0〈/mn〉〈mo stretchy="false"〉}〈/mo〉〈/math〉, there is an orientable closed 3-manifold 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉M〈/mi〉〈/mrow〉〈mrow〉〈mi〉f〈/mi〉〈/mrow〉〈/msub〉〈mo〉=〈/mo〉〈mi〉S〈/mi〉〈mo〉×〈/mo〉〈mi〉I〈/mi〉〈mo stretchy="false"〉/〈/mo〉〈mi〉f〈/mi〉〈/math〉 which is called a mapping torus. It is known that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉M〈/mi〉〈/mrow〉〈mrow〉〈mi〉f〈/mi〉〈/mrow〉〈/msub〉〈/math〉 admits a canonical Heegaard splitting 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉H〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈msub〉〈mrow〉〈mo〉∪〈/mo〉〈/mrow〉〈mrow〉〈mi mathvariant="normal"〉Σ〈/mi〉〈/mrow〉〈/msub〉〈msub〉〈mrow〉〈mi〉H〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈/math〉. By the construction of Namazi [H.Namazi, Topology Appl. 154 (2007), no. 16, 2939-2949], the mapping class group of this Heegaard splitting, denoted by 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si37.gif" overflow="scroll"〉〈mi〉M〈/mi〉〈mi〉o〈/mi〉〈mi〉d〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi mathvariant="normal"〉Σ〈/mi〉〈mo〉;〈/mo〉〈msub〉〈mrow〉〈mi〉H〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mo〉,〈/mo〉〈msub〉〈mrow〉〈mi〉H〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉)〈/mo〉〈/math〉, contains a reducible mapping class which has infinitely order. So it is interesting to know that for a given element in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si37.gif" overflow="scroll"〉〈mi〉M〈/mi〉〈mi〉o〈/mi〉〈mi〉d〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi mathvariant="normal"〉Σ〈/mi〉〈mo〉;〈/mo〉〈msub〉〈mrow〉〈mi〉H〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mo〉,〈/mo〉〈msub〉〈mrow〉〈mi〉H〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉)〈/mo〉〈/math〉, whether it is reducible or not.〈/p〉 〈p〉Using the translation length of 〈em〉f〈/em〉 in the curve complex, we prove that if 〈em〉f〈/em〉 is the identity map or its translation length is at least 8, then each element of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si37.gif" overflow="scroll"〉〈mi〉M〈/mi〉〈mi〉o〈/mi〉〈mi〉d〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi mathvariant="normal"〉Σ〈/mi〉〈mo〉;〈/mo〉〈msub〉〈mrow〉〈mi〉H〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mo〉,〈/mo〉〈msub〉〈mrow〉〈mi〉H〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 is reducible.〈/p〉 〈/div〉

Description: 〈p〉Publication date: Available online 18 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications〈/p〉 〈p〉Author(s): A. Giraldo, M.A. Morón, Á. Sánchez-González〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉This paper is devoted to introducing additional structure on Čech homology groups. First, we redefine the Čech homology groups in terms of what we have called approximative homology by using approximative sequences of cycles, just as Borsuk introduced shape groups using approximative maps. From this point on, we are able to construct complete ultrametrics on Čech homology groups. The uniform type (and then the group topology) generated by the ultrametric leads to a shape invariant which we use to deduce topological consequences.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 18 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications〈/p〉 〈p〉Author(s): Alexander Dranishnikov, James Keesling〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉For every compact metrizable group 〈em〉G〈/em〉 there is a free 〈em〉G〈/em〉-action on the Hilbert space which is universal for free 〈em〉G〈/em〉-actions on compact metric spaces.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 April 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 257〈/p〉 〈p〉Author(s): 〈/p〉

Description: 〈p〉Publication date: 15 May 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 258〈/p〉 〈p〉Author(s): Shouman Das〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper we demonstrate a calculation to find the genus of the hypercube graph 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉Q〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msub〉〈/math〉 using real moment-angle complex 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si157.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi mathvariant="script"〉Z〈/mi〉〈/mrow〉〈mrow〉〈mi mathvariant="script"〉K〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈msup〉〈mrow〉〈mi〉D〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msup〉〈mo〉,〈/mo〉〈msup〉〈mrow〉〈mi〉S〈/mi〉〈/mrow〉〈mrow〉〈mn〉0〈/mn〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.gif" overflow="scroll"〉〈mi mathvariant="script"〉K〈/mi〉〈/math〉 is the boundary of an 〈em〉n〈/em〉-gon. We also calculate an upper bound for the genus of the quotient graph 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si172.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉Q〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉/〈/mo〉〈msub〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msub〉〈/math〉, where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.gif" overflow="scroll"〉〈msub〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msub〉〈/math〉 represents the cyclic group with 〈em〉n〈/em〉 elements.〈/p〉〈/div〉

Description: 〈p〉Publication date: Available online 15 March 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications〈/p〉 〈p〉Author(s): N. Noble〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉A space X for which X〈sup〉〈em〉ω〈/em〉〈/sup〉 is Lindelöf is called powerfully Lindelöf. Extending this term, I call a collection of topological spaces powerfully Lindelöf if the product of each of its countable subcollections is Lindelöf. In 1971 E. Michael showed (CH) that there is no single such class; here I explore what can be said of them. In doing so, I will focus upon some of the techniques used to prove countable products Lindelöf.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 May 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 258〈/p〉 〈p〉Author(s): Liang-Xue Peng, Pei Zhang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉 〈p〉In this article, we study 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/math〉-factorizability (simple 〈em〉sm〈/em〉-factorizability) in paratopological groups. We show that if 〈em〉H〈/em〉 is a continuous open homomorphic image of an 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/math〉-factorizable paratopological group, then 〈em〉H〈/em〉 is 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/math〉-factorizable. This gives an affirmative answer to a problem posed by M. Sanchis and M. Tkachenko in [12, Problem 5.2]. We also prove that every weakly Lindelöf totally 〈em〉ω〈/em〉-narrow paratopological group is simply 〈em〉sm〈/em〉-factorizable.〈/p〉 〈p〉We show that if a paratopological group 〈em〉H〈/em〉 is a continuous homomorphic image of an 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/math〉-factorizable paratopological group and 〈em〉H〈/em〉 is weakly Lindelöf (or first-countable), then 〈em〉H〈/em〉 is simply 〈em〉sm〈/em〉-factorizable. This gives a partial answer to a problem posed by A.V. Arhangel'skii and M. Tkachenko in [1, Problem 7.8].〈/p〉 〈/div〉

Description: 〈p〉Publication date: 15 May 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 258〈/p〉 〈p〉Author(s): Samuel M. Corson〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We present several new theorems concerning the first fundamental group of a path connected metric space. Among the results proven are strengthenings of the main theorems of [23] and [7]. A compactness theorem for the fundamental group of a Peano continuum is given. We also show that a free decomposition of the fundamental group of a locally path connected Polish space cannot be nonconstructive. Numerous other results and examples illustrating the sharpness of our theorems are provided.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 September 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 265〈/p〉 〈p〉Author(s): José G. Anaya, Enrique Castañeda-Alvarado, Alejandro Fuentes-Montes de Oca〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉A connected topological space 〈em〉Z〈/em〉 is unicoherent provided that if 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mi〉Z〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mi〉A〈/mi〉〈mo〉∪〈/mo〉〈mi〉B〈/mi〉〈/math〉 where 〈em〉A〈/em〉 and 〈em〉B〈/em〉 are closed connected subsets of 〈em〉Z〈/em〉, then 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈mi〉A〈/mi〉〈mo〉∩〈/mo〉〈mi〉B〈/mi〉〈/math〉 is connected. Let 〈em〉Z〈/em〉 be a unicoherent topological space and let 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"〉〈mi〉z〈/mi〉〈mo〉∈〈/mo〉〈mi〉Z〈/mi〉〈/math〉, we say that 〈em〉z〈/em〉 makes a hole in 〈em〉Z〈/em〉 if 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"〉〈mi〉Z〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉−〈/mo〉〈mo stretchy="false"〉{〈/mo〉〈mi〉z〈/mi〉〈mo stretchy="false"〉}〈/mo〉〈/math〉 is not unicoherent. Let 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.svg"〉〈mi〉H〈/mi〉〈mi〉S〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mi〉C〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo stretchy="false"〉/〈/mo〉〈msub〉〈mrow〉〈mi〉F〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 be the hyperspace suspension of a continuum 〈em〉X〈/em〉, where 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si120.svg"〉〈mi〉C〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 is the hyperspace of all nonempty subcontinua of 〈em〉X〈/em〉, and let 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si7.svg"〉〈msub〉〈mrow〉〈mi〉F〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 denote the space of singletons of 〈em〉X〈/em〉. In this paper we study the following problem: for which 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si265.svg"〉〈mi〉A〈/mi〉〈mo〉∈〈/mo〉〈mi〉H〈/mi〉〈mi〉S〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉, 〈em〉A〈/em〉 makes a hole in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si136.svg"〉〈mi〉H〈/mi〉〈mi〉S〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉. We present the solution to this problem and we make an application to give the classification theorems for some families of continua.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 September 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 265〈/p〉 〈p〉Author(s): Leijie Wang, Taras Banakh〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉For a Tychonoff space 〈em〉X〈/em〉 and a subspace 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mi〉Y〈/mi〉〈mo〉⊂〈/mo〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/math〉, we study Baire category properties of the space 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si12.svg"〉〈msub〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mo lspace="0em" rspace="0em" stretchy="false"〉↓〈/mo〉〈mi mathvariant="sans-serif"〉F〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo〉,〈/mo〉〈mi〉Y〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 of continuous functions from 〈em〉X〈/em〉 to 〈em〉Y〈/em〉, endowed with the Fell hypograph topology. We characterize pairs 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"〉〈mi〉X〈/mi〉〈mo〉,〈/mo〉〈mi〉Y〈/mi〉〈/math〉 for which the function space 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si12.svg"〉〈msub〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mo lspace="0em" rspace="0em" stretchy="false"〉↓〈/mo〉〈mi mathvariant="sans-serif"〉F〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo〉,〈/mo〉〈mi〉Y〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 is ∞-meager, meager, Baire, Choquet, strong Choquet, (almost) complete-metrizable or (almost) Polish.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 September 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 265〈/p〉 〈p〉Author(s): Donal O'Regan〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper we present a very simple result which will enable us to establish the topological transversality theorem in a general setting.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 September 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 265〈/p〉 〈p〉Author(s): Peter Eliaš〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Given a family of continuous real functions 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mi mathvariant="script"〉G〈/mi〉〈/math〉, let 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈msub〉〈mrow〉〈mi〉R〈/mi〉〈/mrow〉〈mrow〉〈mi mathvariant="script"〉G〈/mi〉〈/mrow〉〈/msub〉〈/math〉 be a binary relation defined as follows: a continuous function 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"〉〈mi〉f〈/mi〉〈mo〉:〈/mo〉〈mi mathvariant="double-struck"〉R〈/mi〉〈mo stretchy="false"〉→〈/mo〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/math〉 is in the relation with a closed set 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"〉〈mi〉E〈/mi〉〈mo〉⊆〈/mo〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/math〉 if and only if there exists 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.svg"〉〈mi〉g〈/mi〉〈mo〉∈〈/mo〉〈mi mathvariant="script"〉G〈/mi〉〈/math〉 such that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.svg"〉〈mi〉f〈/mi〉〈mo stretchy="false"〉↾〈/mo〉〈mi〉E〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mi〉g〈/mi〉〈mo stretchy="false"〉↾〈/mo〉〈mi〉E〈/mi〉〈/math〉. We consider a Galois connection between families of continuous functions and hereditary families of closed sets of reals naturally associated to 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈msub〉〈mrow〉〈mi〉R〈/mi〉〈/mrow〉〈mrow〉〈mi mathvariant="script"〉G〈/mi〉〈/mrow〉〈/msub〉〈/math〉. We study complete lattices determined by this connection and prove several results showing the dependence of the properties of these lattices on the properties of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mi mathvariant="script"〉G〈/mi〉〈/math〉. In some special cases we obtain exact description of these lattices.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 September 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 265〈/p〉 〈p〉Author(s): Jing Zhang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉For a given paratopological group 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mo stretchy="false"〉(〈/mo〉〈mi〉H〈/mi〉〈mo〉,〈/mo〉〈mi〉τ〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉, the quasitopological group 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈msub〉〈mrow〉〈mi〉Q〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉H〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 with a weaker topology than 〈em〉τ〈/em〉 were investigated and some problems were posed by I. Sánchez and M. Tkachenko. In this note, we give partial answers to two of the problems. Some cardinal inequalities are obtained: (1) if 〈em〉H〈/em〉 is a quasitopological group, then 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si216.svg"〉〈mi〉i〈/mi〉〈mi〉b〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉H〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo〉≤〈/mo〉〈mi〉e〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉H〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉; (2) if 〈em〉H〈/em〉 is a paratopological group, then 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si242.svg"〉〈mi〉i〈/mi〉〈mi〉b〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈msub〉〈mrow〉〈mi〉Q〈/mi〉〈/mrow〉〈mrow〉〈mn〉3〈/mn〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉H〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo stretchy="false"〉)〈/mo〉〈mo〉≤〈/mo〉〈mi〉w〈/mi〉〈mi〉l〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉H〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉; (3) if 〈em〉H〈/em〉 is a saturated semitopological group, then 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si284.svg"〉〈mi〉i〈/mi〉〈mi〉b〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉H〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo〉≤〈/mo〉〈mi〉s〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉H〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉; (4) if 〈em〉H〈/em〉 is a semitopological group, then 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si315.svg"〉〈mi〉i〈/mi〉〈mi〉b〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈msub〉〈mrow〉〈mi〉Q〈/mi〉〈/mrow〉〈mrow〉〈mn〉3〈/mn〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉H〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo stretchy="false"〉)〈/mo〉〈mo〉≤〈/mo〉〈mi〉c〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉H〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 September 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 265〈/p〉 〈p〉Author(s): Steven Clontz, Jared Holshouser〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We relate the property of discrete selectivity and its corresponding game, both recently introduced by V.V. Tkachuk, to a variety of selection principles and point-picking games. In particular we show that player II can win the discrete selectivity game on 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 if and only if player II can win a variant of the point-open game on 〈em〉X〈/em〉. We also show that the existence of limited information strategies in the discrete selectivity game on 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi〉C〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉(〈/mo〉〈mi〉X〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 for either player are equivalent to other well-known topological properties.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 September 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 265〈/p〉 〈p〉Author(s): Li-Hong Xie, Peng-Fei Yan〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, the class of weakly 〈em〉ω〈/em〉-admissible paratopological groups is introduced. We mainly show that: (1) Every bounded set in a weakly 〈em〉ω〈/em〉-admissible paratopological group is 〈em〉p〈/em〉-bounded for each free ultrafilter 〈em〉p〈/em〉 on 〈em〉ω〈/em〉; (2) every 〈em〉C〈/em〉-compact set in a weakly 〈em〉ω〈/em〉-admissible paratopological group is strongly 〈em〉r〈/em〉-pseudocompact. The class of weakly 〈em〉ω〈/em〉-admissible paratopological groups contains all totally 〈em〉ω〈/em〉-narrow, precompact, locally feebly compact, weakly Lindelöf and saturated, and 〈em〉ω〈/em〉-admissible paratopological groups. It is worth mentioning that all the aforementioned spaces are not to satisfy any separation axiom. Some Sanchis, Sánchez and Tkachenko's results are improved.〈/p〉〈/div〉

Description: 〈p〉Publication date: 15 September 2019〈/p〉 〈p〉〈b〉Source:〈/b〉 Topology and its Applications, Volume 265〈/p〉 〈p〉Author(s): Toshimichi Usuba〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We construct a normal countably tight 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi〉T〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈/math〉 space 〈em〉X〈/em〉 with 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si19.svg"〉〈mi〉t〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈msub〉〈mrow〉〈mi〉X〈/mi〉〈/mrow〉〈mrow〉〈mi〉δ〈/mi〉〈/mrow〉〈/msub〉〈mo stretchy="false"〉)〈/mo〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈msup〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈mrow〉〈mi〉ω〈/mi〉〈/mrow〉〈/msup〉〈/math〉. This is an answer to the question posed by Dow-Juhász-Soukup-Szentmiklóssy-Weiss [5]. We also show that if the continuum is not so large, then the tightness of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"〉〈msub〉〈mrow〉〈mi〉G〈/mi〉〈/mrow〉〈mrow〉〈mi〉δ〈/mi〉〈/mrow〉〈/msub〉〈/math〉-modifications of countably tight spaces can be arbitrary large up to the least 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si21.svg"〉〈msub〉〈mrow〉〈mi〉ω〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈/math〉-strongly compact cardinal.〈/p〉〈/div〉