\sigma-Point-Discrete cs^{\ast}-Networks and wcs^{\ast}-Networks

Let $\mathcal{B}=\{B_{\alpha}:\alpha\in H\}$ be a family of subsets of a space $X$. $\mathcal{B}$ is {\it point-discrete} (or {\it weakly hereditarily closure-preserving}) if $\{x_{\alpha}:\alpha\in H\}$ is closed discrete in $X$, whenever $x_{\alpha}\in B_{\alpha}$ for each $\alpha\in H$. In this paper, we mainly discuss the spaces with a $\sigma$-point-discrete $cs^{\ast}$-network or $wcs^{\ast}$-network, and the main results are that (1)If $X$ has a $\sigma$-point-discrete $cs^{\ast}$-network and $\sigma X$ contains no closed copy of $S_{\omega}$, then $X$ has a $\sigma$-compact-finite $cs^{\ast}$-network; (2) $S_{\omega_{1}}\times S_{1}$ has a $\sigma$--point-discrete $cs^{\ast}$-network.