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

denotes the real-valued continuous functions on

Y\subseteq X Y ⊆ X

having continuous extensions to a Tychonoff space

X X

, with pointwise topology inherited from

C_{p}(Y) C p ( Y )

. We recently proved

C_{p}(Y) C p ( Y )

is distinguished

\Leftrightarrow ⇔

it is a large subspace of

\mathbb{R}^{Y} R Y

. We prove

C_{p}\left( Y|X\right) C p ( Y ∣ X )

is always a large subspace of

C_{p}(Y) C p ( Y )

. Thus

C_{p}\left( Y|X\right) C p ( Y ∣ X )

is always quasibarrelled; always has a feral strong dual; has a quasibarrelled countable enlargement

\Leftrightarrow ⇔

Y Y

is infinite; is distinguished

\Leftrightarrow ⇔

C_{p}(Y) C p ( Y )

is distinguished; is a Montel space

\Leftrightarrow ⇔

Y Y

is discrete and

C C

-embedded in

X X

. ‘Nice’ countable covers for

C_{p}\left( Y|X\right) C p ( Y ∣ X )

yield potent summary theorems that solve open problems, characterize

P P

-spaces anew, and complete the list of Velichko variations. For example, Summary III: Assume

Y Y

is dense in

X X

.

Y Y

is a

P P

-space, or

X X

is pseudocompact, or both

\Leftrightarrow ⇔

C_{p}\left( Y|X\right) C p ( Y ∣ X )

is countably covered by sets that are, respectively, relatively sequentially complete in

C_{p}(Y) C p ( Y )

, or bounded, or both. Putting

Y=X Y = X

, one quickly comprehends Velichko variations à la Arkhangel'skiĭ.