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

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 $\mathcal F_n(X)$