Metrizable compacta in the space of continuous functions with the topology of pointwise convergence

Abstract

We prove that every point-finite family of nonempty functionally open sets in a topological space $X$ has the cardinality at most an infinite cardinal $\kappa$ if and only if $w(X)\leq\kappa$ for every Valdivia compact space $Y\subseteq C_p(X)$. Correspondingly a Valdivia compact space $Y$ has the weight at most an infinite cardinal $\kappa$ if and only if every point-finite family of nonempty open sets in $C_p(Y)$ has the cardinality at most $\kappa$, that is $p(C_p(Y))\leq \kappa$. Besides, it was proved that $w(Y)=p(C_p(Y))$ for every linearly ordered compact $Y$. In particular, a Valdivia compact space or linearly ordered compact space $Y$ is metrizable if and only if $p(C_p(Y))=\aleph_0$. This gives answer to a question of O.~Okunev and V.~Tkachuk.