ISSN:
1436-5081
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract This paper presents a general investigation of the relations between structural properties of a totally ordered abelian semigroupS and the properties of various “topological” structures, such as topologies, bitopologies and (semi-)uniformities on a spaceX induced byS-valued distance functionsd∶X×X→S satisfyingd(x,y)=0 iffx=y and the triangular inequalityd(x,z)≤d(x,y)+d(y,z), for allx,y,z∈X. Since a linearly ordered abelian semigroupS need not be a topological semigroup with respect to its order topology we have to consider two cases: the case where addition inS is continuous at 0∈S, and the case where it is not. For both cases, we state several metrization theorems, examples and applications. In this connection, we are also concerned with some special basis-properties of topological spaces. Closely connected is the program stated byAlexandroff-Bourbaki (amongst others) to investigate to what extent countability inherent in matrization theory can be replaced by order-theoretic properties.—Distinguishing between symmetric and not necessarily symmetric distancesd S we obtain a theory containing the theory ofω µ -metrics andω µ- quasimetrics.—As far as it concerns not necessarily symmetric distancesd onX, it seems adequate to study the bitopological structure (τ e ,τ r ) induced onX byd and the “inverse” distanced −1 respectively. This is done in § 4 where, in this respect, we also generalize a well-known theorem ofSion andZelmer.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01541638
