The Ever Large Subspace C_p(Y|X): Distinguished, Montel, Covered Nicely?

$C_{p}\left( Y|X\right)$ denotes the real-valued continuous functions on $Y\subseteq X$ having continuous extensions to a Tychonoff space $X$, with pointwise topology inherited from $C_{p}(Y)$. We recently proved $C_{p}(Y)$ is distinguished $\Leftrightarrow$ it is a large subspace of $\mathbb{R}^{Y}$. We prove $C_{p}\left( Y|X\right)$ is always a large subspace of $C_{p}(Y)$. Thus $C_{p}\left( Y|X\right)$ is always quasibarrelled; always has a feral strong dual; has a quasibarrelled countable enlargement $\Leftrightarrow$ $Y$ is infinite; is distinguished $\Leftrightarrow$ $C_{p}(Y)$ is distinguished; is a Montel space $\Leftrightarrow$ $Y$ is discrete and $C$-embedded in $X$. `Nice' countable covers for $C_{p}\left( Y|X\right)$ yield potent summary theorems that solve open problems, characterize $P$-spaces anew, and complete the list of Velichko variations. For example, Summary III: Assume $Y$ is dense in $X$. $Y$ is a $P$-space, or $X$ is pseudocompact, or both $\Leftrightarrow$ $C_{p}\left( Y|X\right)$ is countably covered by sets that are, respectively, relatively sequentially complete in $C_{p}(Y)$, or bounded, or both. Putting $Y=X$, one quickly comprehends Velichko variations à la Arkhangel'skiĭ.