Scholar - SciOpen

Let $X$ be an infinite Tychonoff space, and $Y$ be a topological subspace of $X$ . In this paper, we study some covering properties of the subspace $C_{p} (Y | X)$ of $C_{p} (Y)$ consisting of those functions $f \in C (Y)$ which admit a continuous extension to $X$ equipped with the relative topology of $C_{p} (Y)$ . Among other results, we show that $(i)$ $C_{p} (Y | X)$ has a fundamental bounded resolution if and only if $Y$ is countable; when $X$ is realcompact and $Y$ is closed in $X$ , we have $(i i)$ if $C_{p} (Y | X)$ admits a resolution of convex compact sets that swallows the local null sequences in $C_{p} (Y | X)$ , then $Y$ is countable and discrete; $(i i i)$ if $C_{p} (Y | X)$ admits a compact resolution that swallows the compact sets, then $Y$ is also countable and discrete, and, as a corollary, we deduce that $C_{p} (Y | X)$ admits a compact resolution that swallows the compact sets if and only if $C_{p} (Y | X)$ is a Polish space. We also prove that $(i v)$ for a metrizable space $X$ , $C_{p} (X)$ is a quasi- $(L B)$ -space if and only if $X$ is $σ$ -compact, and hence for a subspace $Y$ of $X$ , the space $C_{p} (Y | X)$ is a quasi- $(L B)$ -space. We include some examples and observations that answer natural questions raised in this paper.