Some applications of discrete selectivity and Banakh property in function spaces

Abstract

We establish that an uncountable space X must be essentially uncountable whenever its extent and tightness are countable. As a consequence, the equality $\mathrm{ext}(X)= t(X)=\omega$ implies that the space $C_{p}(X, [0,1])$ is discretely selective. If X is a metrizable space, then $C_{p}(X, [0,1])$ has the Banakh property if and only if so does $C_{p}(Y, [0,1])$ for some closed separable $Y\subset X$. We apply the above results to show that, for a metrizable X, the space $C_{p}(X, [0,1])$ is strongly dominated by a second countable space if and only if X is homeomorphic to $D\,{\oplus }\, M$ where D is a discrete space and M is countable. For a metrizable space X, we also prove that $C_{p}(X,[0,1])$ has the Lindelöf $\Sigma$-property if and only if the set of non-isolated points of X is second countable. Our results solve several open questions.