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.
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McCoy, R.A. Densely Continuous Forms in Vietoris Hyperspaces. Set-Valued Analysis 8, 267–271 (2000). https://doi.org/10.1023/A:1008773312587
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DOI: https://doi.org/10.1023/A:1008773312587