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

TL;DR

This paper characterizes when Valdivia and linearly ordered compact spaces are metrizable based on the point-finite families in the space of continuous functions with pointwise convergence topology, answering a question by Okunev and Tkachuk.

Contribution

It establishes a new criterion linking the weight of Valdivia compacta and linear orders to point-finite families in function spaces, providing a metrization characterization.

Findings

A Valdivia compact space has weight at most κ iff every point-finite family in Cp(X) has size ≤ κ.

For linearly ordered compact Y, w(Y) equals p(Cp(Y)).

Y is metrizable iff p(Cp(Y))=ℵ₀.

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.