If K is a Valdivia compact space, then $C_{\hspace{-1.111pt}p}(K)$ is uniformly $\psi$-separable

Abstract

We prove that, for any countably compact subspace X of a $\Sigma$-product of real lines, the space $C_{\hspace{-1.111pt}p}(X)$ is uniformly $\psi$-separable, that is, has a uniformly dense subset of countable pseudocharacter. This result implies that $C_{\hspace{-1.111pt}p}(K)$ is uniformly $\psi$-separable whenever K is a Valdivia compact space. We show that the existence of a uniformly dense realcompact subset of $C_{\hspace{-1.111pt}p}(X)$ need not imply that $C_{\hspace{-1.111pt}p}(X)$ is realcompact even if the space X is compact. We also establish that $C_{\hspace{-1.111pt}p}(X)$ can fail to be $\omega$-monolithic if it has a uniformly dense $\omega$-monolithic subspace. Furthermore, an example is given of spaces X and Y such that both $C_{\hspace{-1.111pt}p}(X)$ and $C_{\hspace{-1.111pt}p}(Y)$ are Lindelöf but $C_{\hspace{-1.111pt}p}(X\,{\times }\, Y)$ has no uniformly dense Lindelöf subspace.