Vietoris topology on spaces dominated by second countable ones

Author:

Islas Carlos1,Jardon Daniel2

Affiliation:

1. 1Academia de Matemáticas, Universidad Autónoma de la Ciudad de México, San Lorenzo 290 Colonia del Valle Sur, CP 03100, Mexico City, Mexico

2. 2Academia de Matemáticas, Universidad Autónoma de la Ciudad de México, Calzada Ermita Iztapalapa 4163 Colonia Lomas de Zaragoza, CP 09620, Mexico City, Mexico

Abstract

AbstractFor a given space X let C(X) be the family of all compact subsets of X. A space X is dominated by a space M if X has an M-ordered compact cover, this means that there exists a family F = {FK : K ∈ C(M)} ⊂ C(X) such that ∪ F = X and K ⊂ L implies that FK ⊂ FL for any K;L ∈ C(M). A space X is strongly dominated by a space M if there exists an M-ordered compact cover F such that for any compact K ⊂ X there is F ∈ F such that K ⊂ F . Let K(X) D C(X)\{Ø} be the set of all nonempty compact subsets of a space X endowed with the Vietoris topology. We prove that a space X is strongly dominated by a space M if and only if K(X) is strongly dominated by M and an example is given of a σ-compact space X such that K(X) is not Lindelöf†. It is stablished that if the weight of a scattered compact space X is not less than c, then the spaces Cp(K(X)) and K(Cp(X)) are not Lindelöf Σ. We show that if X is the one-point compactification of a discrete space, then the hyperspace K(X) is semi-Eberlein compact.

Publisher

Walter de Gruyter GmbH

Subject

General Mathematics

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1. The hyperspace of a semi-Eberlein compact space is semi-Eberlein;Topology and its Applications;2023-04

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