## Luong Q. Tuyen, Ong V. Tuyen

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

Comment.Math.Univ.Carolin. 61,2 (2020) 257-263.**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 $\mathcal F_n(X)$.

**Keywords:** $\sigma$-$(P)$-property; $cn$-network; $ck$-network; strict $\sigma$-space; strict $\aleph$-space

**DOI:** DOI 10.14712/1213-7243.2020.025

**AMS Subject Classification:** 54B20 54D20

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