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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

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Vietoris topology on spaces dominated by second countable ones

Carlos Islas
  • Academia de Matemáticas, Universidad Autónoma de la Ciudad de México, San Lorenzo 290 Colonia del Valle Sur, CP 03100, Mexico City, Mexico
  • Email:
/ Daniel Jardon
  • Academia de Matemáticas, Universidad Autónoma de la Ciudad de México, Calzada Ermita Iztapalapa 4163 Colonia Lomas de Zaragoza, CP 09620, Mexico City, Mexico
  • Email:
Published Online: 2015-01-07 | DOI: https://doi.org/10.1515/math-2015-0018


For 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.

Keywords : Strong domination by second countable spaces; Hemicompact space; Lindelöf p-space; Lindelöf -space; Vietoris topology; One-point compactification; Eberlein compact; Scattered spaces


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About the article

Received: 2014-08-14

Accepted: 2014-12-06

Published Online: 2015-01-07

Citation Information: Open Mathematics, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2015-0018. Export Citation

© 2015 Carlos Islas and Daniel Jardon. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. (CC BY-NC-ND 3.0)

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