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- W2020166407 abstract "We investigate the problem of characterizing those quasi-uniform spaces for which the Vietoris topology is compatible with the Hausdorff quasi-uniformity. In particular, we prove that the Vietoris topology of a quasi-uniform space ( X , u ) is compatible with the Hausdorff quasi-uniformity on the family K 0 ( X ) of nonempty compact subsets of X , if and only if for each K ∈ K 0 ( X ) , u - 1 | K is precompact. We show that for a T 1 quasi-uniform space ( X , u ) , the Vietoris topology is compatible with the Hausdorff quasi-uniformity on the family of nonempty closed subsets of X , if and only if u is equinormal and u - 1 is hereditarily precompact. We also discuss the problem in the setting of quasi-metric spaces and show, among other results, that the Vietoris topology of a quasi-metric space ( X , d ) is compatible with the Hausdorff extended quasi-pseudo-metric on the family of nonempty subsets of X , if and only if the quasi-uniformity induced by d coincides with the Pervin quasi-uniformity of X ." @default.
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- W2020166407 date "2002-10-01" @default.
- W2020166407 modified "2023-09-29" @default.
- W2020166407 title "The relationship between the Vietoris topology and the Hausdorff quasi-uniformity" @default.
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- W2020166407 doi "https://doi.org/10.1016/s0166-8641(01)00252-8" @default.
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