Covering properties of $C_p\left(Y|X\right)$

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 $C_p\left(Y|X\right)$ of $C_p\left(Y\right)$ consisting of those functions $f\in C\left(Y\right)$ which admit a continuous extension to $X$ equipped with the relative topology of $C_p\left(Y\right)$. Among other results, we show that $\left(i\right)$ $C_p(Y|X)$ has a fundamental bounded resolution if and only if $Y$ is countable; when $X$ is realcompact and $Y$ is closed in $X$, we have $\left(ii\right)$ 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; $\left(iii\right)$ 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 $\left(iv\right)$ for a metrizable space $X$, $C_p\left(X\right)$ is a quasi- $\left(LB\right)$-space if and only if $X$ is $\sigma$-compact, and hence for a subspace $Y$ of $X$, the space $C_p\left(Y|X\right)$ is a quasi- $\left(LB\right)$-space. We include some examples and observations that answer natural questions raised in this paper.