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 Title: On the $n$-fold symmetric product of a space with a $\sigma$-$(P)$-property $cn$-network ($ck$-network) (English) Author: Tuyen, Luong Q. Author: Tuyen, Ong V. Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 61 Issue: 2 Year: 2020 Pages: 257-263 Summary lang: English . Category: math . Summary: 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 $\mathcal F_n(X)$. (English) Keyword: $\sigma$-$(P)$-property Keyword: $cn$-network Keyword: $ck$-network Keyword: strict $\sigma$-space Keyword: strict $\aleph$-space MSC: 54B20 MSC: 54D20 idZBL: Zbl 07286004 idMR: MR4143708 DOI: 10.14712/1213-7243.2020.025 . Date available: 2020-10-13T13:15:53Z Last updated: 2021-02-25 Stable URL: http://hdl.handle.net/10338.dmlcz/148281 . Reference: [1] Borsuk K., Ulam S.: On symmetric products of topological spaces.Bull. Amer. Math. Soc. 37 (1931), no. 12, 875–882. Zbl 0003.22402, MR 1562283, 10.1090/S0002-9904-1931-05290-3 Reference: [2] Gabriyelyan S. S., Kakol J.: On $\mathfrak{P}$-spaces and related concepts.Topology Appl. 191 (2015), 178–198. MR 3361065 Reference: [3] Good C., Macías S.: Symmetric products of generalized metric spaces.Topology Appl. 206 (2016), 93–114. MR 3494434 Reference: [4] Peng L.-X., Sun Y.: A study on symmetric products of generalized metric spaces.Topology Appl. 231 (2017), 411–429. MR 3712980, 10.1016/j.topol.2017.09.036 Reference: [5] Tang Z., Lin S., Lin F.: Symmetric products and closed finite-to-one mappings.Topology Appl. 234 (2018), 26–45. MR 3739454, 10.1016/j.topol.2017.11.004 .

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