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 $$\ $Y$\ is infinite; is distinguished $$\ $C_{p}(Y)$\ 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 $$\ $C_{p}\left( Y|X\right)$\ 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 \`a la Arkhangel'ski\u{\i}.