On a Question of Closed Maps of S. Lin

Let X be a regular $T_1$-space such that each single point set is a $G_{\delta}$ set. Denot 'hereditarily closure-preserving' by 'HCP'. To consider a question of closed maps of S. Lin in [6], we improve some results of Foged in [1], and prove the following propositions. Proposition 1. $D\;=\;\{x{\in}X\;:\;\mid\{F{\in}\cal{F}:x{\in}F\}\mid{\geq}{\aleph}_0\}$ is discrete and closed if $\cal{F}$ is a collection of HCP. Proposition 2. $\cal{H}\;=\;\{{\cup}\cal{F}'\;:\;F'$ is an fininte subcolletion of $\cal{F}_n\}$ is HCP if $\cal{F}$ is a collection of HCP. Proposition 3. Let (X,$\tau$) have a $\sigma$-HCP k-network. Then (X,$\tau$) has a $\sigma$-HCP k-network F = ${\cup}_n\cal{F}_n$ such that such tat: (i) $\cal{F}_n\;\subset\;\cal{F}_{n+1}$, (ii) $D_n\;=\;\{x{\in}X\;:\;\mid\{F{\in}\cal{F}_n\;:\;x{\in}F\}\mid\;{\geq}\;{\aleph}_0\}$ is a discrete closed set and (iii) each $\cal{F}_n$ is closed to finite intersections.