About $wcs$-covers and $wcs^*$-networks on the Vietoris hyperspace $\mathcal F(X)$
Commentationes Mathematicae Universitatis Carolinae, Tome 64 (2023) no. 4, pp. 519-527
We study some generalized metric properties on the hyperspace $\mathcal F(X)$ of finite subsets of a space $X$ endowed with the Vietoris topology. We prove that $X$ has a point-star network consisting of (countable) $wcs$-covers if and only if so does $\mathcal F(X)$. Moreover, $X$ has a sequence of $wcs$-covers with property $(P)$ which is a point-star network if and only if so does $\mathcal F(X)$, where $(P)$ is one of the following properties: point-finite, point-countable, compact-finite, compact-countable, locally finite, locally countable. On the other hand, $X$ has a $wcs^*$-network with property $\sigma$-$(P)$ if and only if so does $\mathcal F(X)$. By these results, we obtain some results related to the images of metric spaces and separable metric spaces under some kinds of continuous mappings on the Vietoris hyperspace $\mathcal F(X)$.
We study some generalized metric properties on the hyperspace $\mathcal F(X)$ of finite subsets of a space $X$ endowed with the Vietoris topology. We prove that $X$ has a point-star network consisting of (countable) $wcs$-covers if and only if so does $\mathcal F(X)$. Moreover, $X$ has a sequence of $wcs$-covers with property $(P)$ which is a point-star network if and only if so does $\mathcal F(X)$, where $(P)$ is one of the following properties: point-finite, point-countable, compact-finite, compact-countable, locally finite, locally countable. On the other hand, $X$ has a $wcs^*$-network with property $\sigma$-$(P)$ if and only if so does $\mathcal F(X)$. By these results, we obtain some results related to the images of metric spaces and separable metric spaces under some kinds of continuous mappings on the Vietoris hyperspace $\mathcal F(X)$.
DOI :
10.14712/1213-7243.2024.011
Classification :
54B20, 54C10, 54D20, 54E40
Keywords: hyperspace; generalized metric property; $wcs$-cover; $wcs^*$-network
Keywords: hyperspace; generalized metric property; $wcs$-cover; $wcs^*$-network
@article{10_14712_1213_7243_2024_011,
author = {Tuyen, Luong Q. and Tuyen, Ong V. and Tuan, Phan D. and Truc, Nguzen X.},
title = {About $wcs$-covers and $wcs^*$-networks on the {Vietoris} hyperspace $\mathcal F(X)$},
journal = {Commentationes Mathematicae Universitatis Carolinae},
pages = {519--527},
year = {2023},
volume = {64},
number = {4},
doi = {10.14712/1213-7243.2024.011},
mrnumber = {4813801},
zbl = {07953697},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.14712/1213-7243.2024.011/}
}
TY - JOUR AU - Tuyen, Luong Q. AU - Tuyen, Ong V. AU - Tuan, Phan D. AU - Truc, Nguzen X. TI - About $wcs$-covers and $wcs^*$-networks on the Vietoris hyperspace $\mathcal F(X)$ JO - Commentationes Mathematicae Universitatis Carolinae PY - 2023 SP - 519 EP - 527 VL - 64 IS - 4 UR - http://geodesic.mathdoc.fr/articles/10.14712/1213-7243.2024.011/ DO - 10.14712/1213-7243.2024.011 LA - en ID - 10_14712_1213_7243_2024_011 ER -
%0 Journal Article %A Tuyen, Luong Q. %A Tuyen, Ong V. %A Tuan, Phan D. %A Truc, Nguzen X. %T About $wcs$-covers and $wcs^*$-networks on the Vietoris hyperspace $\mathcal F(X)$ %J Commentationes Mathematicae Universitatis Carolinae %D 2023 %P 519-527 %V 64 %N 4 %U http://geodesic.mathdoc.fr/articles/10.14712/1213-7243.2024.011/ %R 10.14712/1213-7243.2024.011 %G en %F 10_14712_1213_7243_2024_011
Tuyen, Luong Q.; Tuyen, Ong V.; Tuan, Phan D.; Truc, Nguzen X. About $wcs$-covers and $wcs^*$-networks on the Vietoris hyperspace $\mathcal F(X)$. Commentationes Mathematicae Universitatis Carolinae, Tome 64 (2023) no. 4, pp. 519-527. doi: 10.14712/1213-7243.2024.011
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