Discrete reflexivity in function spaces

Abstract

We study systematically when $C_p(X)$ has a topological property $\mathcal{P}$ if $C_p(X)$ is discretely $\mathcal{P}$, i.e., the set $\overline {D}$ has $\mathcal{P}$ for every discrete subspace $D\subset C_p(X)$. We prove that it is independent of ZFC whether discrete metrizability of $C_p(X)$ implies its metrizability for a compact space $X$. We show that it is consistent with ZFC that countable tightness and Lindelöf $\Sigma$-property are not discretely reflexive in spaces $C_p(X)$. It is also established that a space $X$ must be countable and discrete if $C_p(X)$ is discretely Čech-complete. If $C_p(X)$ is discretely $\sigma$-compact then $X$ has to be finite.