On the $n$-fold symmetric product of a space with a $\sigma$-$\left(P\right)$-property $cn$-network ($ck$-network)

Luong Q. Tuyen; Ong V. Tuyen

Commentationes Mathematicae Universitatis Carolinae (2020)

• Volume: 61, Issue: 2, page 257-263
• ISSN: 0010-2628

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Abstract

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We study the relation between a space $X$ satisfying certain generalized metric properties and its $n$-fold symmetric product ${ℱ}_n\left(X\right)$ satisfying the same properties. We prove that $X$ has a $\sigma$-$\left(P\right)$-property $cn$-network if and only if so does ${ℱ}_n\left(X\right)$. Moreover, if $X$ is regular then $X$ has a $\sigma$-$\left(P\right)$-property $ck$-network if and only if so does ${ℱ}_n\left(X\right)$. By these results, we obtain that $X$ is strict $\sigma$-space (strict $\aleph$-space) if and only if so is ${ℱ}_n\left(X\right)$.

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Tuyen, Luong Q., and Tuyen, Ong V.. "On the $n$-fold symmetric product of a space with a $\sigma $-$(P)$-property $cn$-network ($ck$-network)." Commentationes Mathematicae Universitatis Carolinae 61.2 (2020): 257-263. <http://eudml.org/doc/296938>.

@article{Tuyen2020,
abstract = {We study the relation between a space $X$ satisfying certain generalized metric properties and its $n$-fold symmetric product $\mathcal \{F\}_n(X)$ satisfying the same properties. We prove that $X$ has a $\sigma $-$(P)$-property $cn$-network if and only if so does $\,\mathcal \{F\}_n(X)$. Moreover, if $\,X$ is regular then $X$ has a $\sigma $-$(P)$-property $ck$-network if and only if so does $\,\mathcal \{F\}_n(X)$. By these results, we obtain that $X$ is strict $\sigma $-space (strict $\aleph $-space) if and only if so is $\mathcal \{F\}_n(X)$.},
author = {Tuyen, Luong Q., Tuyen, Ong V.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {$\sigma $-$(P)$-property; $cn$-network; $ck$-network; strict $\sigma $-space; strict $\aleph $-space},
language = {eng},
number = {2},
pages = {257-263},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On the $n$-fold symmetric product of a space with a $\sigma $-$(P)$-property $cn$-network ($ck$-network)},
url = {http://eudml.org/doc/296938},
volume = {61},
year = {2020},
}

TY - JOUR
AU - Tuyen, Luong Q.
AU - Tuyen, Ong V.
TI - On the $n$-fold symmetric product of a space with a $\sigma $-$(P)$-property $cn$-network ($ck$-network)
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2020
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 61
IS - 2
SP - 257
EP - 263
AB - We study the relation between a space $X$ satisfying certain generalized metric properties and its $n$-fold symmetric product $\mathcal {F}_n(X)$ satisfying the same properties. We prove that $X$ has a $\sigma $-$(P)$-property $cn$-network if and only if so does $\,\mathcal {F}_n(X)$. Moreover, if $\,X$ is regular then $X$ has a $\sigma $-$(P)$-property $ck$-network if and only if so does $\,\mathcal {F}_n(X)$. By these results, we obtain that $X$ is strict $\sigma $-space (strict $\aleph $-space) if and only if so is $\mathcal {F}_n(X)$.
LA - eng
KW - $\sigma $-$(P)$-property; $cn$-network; $ck$-network; strict $\sigma $-space; strict $\aleph $-space
UR - http://eudml.org/doc/296938
ER -

References

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1. Borsuk K., Ulam S., 10.1090/S0002-9904-1931-05290-3, Bull. Amer. Math. Soc. 37 (1931), no. 12, 875–882. Zbl0003.22402MR1562283DOI10.1090/S0002-9904-1931-05290-3
2. Gabriyelyan S. S., Kakol J., On $𝔓$-spaces and related concepts, Topology Appl. 191 (2015), 178–198. MR3361065
3. Good C., Macías S., Symmetric products of generalized metric spaces, Topology Appl. 206 (2016), 93–114. MR3494434
4. Peng L.-X., Sun Y., 10.1016/j.topol.2017.09.036, Topology Appl. 231 (2017), 411–429. MR3712980DOI10.1016/j.topol.2017.09.036
5. Tang Z., Lin S., Lin F., 10.1016/j.topol.2017.11.004, Topology Appl. 234 (2018), 26–45. MR3739454DOI10.1016/j.topol.2017.11.004

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