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  • countably paracompact normal spaces  (1)
  • densely continuous form  (1)
  • 1
    Electronic Resource
    Electronic Resource
    Springer
    Set-valued analysis 5 (1997), S. 247-266 
    ISSN: 1572-932X
    Keywords: Heine–Borel property ; densely continuous form ; densely equicontinuous ; densely pointwise bounded
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The set C(X,Y) of continuous functions from a topological space X into a topological space Y is extended to the set D(X,Y) of densely continuous forms from X to Y, such form being a kind of multifunction from X to Y. The topologies of pointwise convergence, uniform convergence, and uniform convergence on compact sets are defined for D(X,Y), for locally compact spaces X and metric spaces Y having a metric satisfying the Heine–Borel property. Under these assumptions, D(X,Y) with the uniform topology is shown to be completely metrizable. In addition, if X is σ compact, D(X,Y) is completely metrizable under the topology of uniform convergence on compact sets. For this latter topology, an Ascoli theorem is established giving necessary and sufficient conditions for a subset of D(X,Y) to be compact.
    Type of Medium: Electronic Resource
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  • 2
    Electronic Resource
    Electronic Resource
    Springer
    Set-valued analysis 8 (2000), S. 267-271 
    ISSN: 1572-932X
    Keywords: densely continuous forms ; Vietoris hyperspaces ; semicontinuous functions ; countably paracompact normal spaces ; locally compact separable metric spaces
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract For countably paracompact normal spaces X and locally compact separable metric spaces Y, a characterization is given for the closure of the set of densely continuous forms from X to Y in the hyperspace of nonempty closed subsets of X × Y under the Vietoris topology. This shows that for such X having no isolated points, every closed subset of X × R that is dense over X can be ‘Vietoris approximated’ by a semicontinuous function on X.
    Type of Medium: Electronic Resource
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