Vietoris topology on spaces dominated by second countable ones

Carlos Islas; Daniel Jardon

Open Mathematics (2015)

  • Volume: 13, Issue: 1, page 188-195, electronic only
  • ISSN: 2391-5455

Abstract

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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){Øbe 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.

How to cite

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Carlos Islas, and Daniel Jardon. "Vietoris topology on spaces dominated by second countable ones." Open Mathematics 13.1 (2015): 188-195, electronic only. <http://eudml.org/doc/268936>.

@article{CarlosIslas2015,
abstract = {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)\{Øbe 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. },
author = {Carlos Islas, Daniel Jardon},
journal = {Open Mathematics},
keywords = {Strong domination by second countable spaces; Hemicompact space; Lindelöf p-space; Lindelöf -space; Vietoris topology; One-point compactification; Eberlein compact; Scattered spaces; strong domination by second countable spaces; hemicompact space; Lindelöf-space; one-point compactification; scattered spaces},
language = {eng},
number = {1},
pages = {188-195, electronic only},
title = {Vietoris topology on spaces dominated by second countable ones},
url = {http://eudml.org/doc/268936},
volume = {13},
year = {2015},
}

TY - JOUR
AU - Carlos Islas
AU - Daniel Jardon
TI - Vietoris topology on spaces dominated by second countable ones
JO - Open Mathematics
PY - 2015
VL - 13
IS - 1
SP - 188
EP - 195, electronic only
AB - 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){Øbe 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.
LA - eng
KW - Strong domination by second countable spaces; Hemicompact space; Lindelöf p-space; Lindelöf -space; Vietoris topology; One-point compactification; Eberlein compact; Scattered spaces; strong domination by second countable spaces; hemicompact space; Lindelöf-space; one-point compactification; scattered spaces
UR - http://eudml.org/doc/268936
ER -

References

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