Function spaces jointly metrizable on compacta (Q493837)

A topological space \(X\) is jointly metrizable on compacta (a JCM space) if there exists a metric \(d\) on \(X\) such that the topology generated by \(d\) restricted to \(K\) coincides with the subspace topology on \(K\) whenever \(K\) is a compact subspace of \(X\). For a space \(X\), let \(C_{p}(X)\) be the set of all real-valued continuous functions on \(X\) endowed with the topology of pointwise convergence. The author studies when \(C_{p}(X)\) has the JCM property. The main results are: (1) If \(C_p(X)\) is a JCM space then the point-finite cellularity of \(X\) is countable and hence \(X\) has the Souslin property. (2) The JCM property of \(C_p(X)\) need not imply that \(\omega_1\) is a caliber of \(X\). (3) If \(X\) is either submetrizable or a P-space then \(C_p(C_p(X))\) is jointly metrizable on compacta. (4) If \(X\) is a dyadic compact space, then \(C_p(X)\) is jointly metrizable on compacta. Therefore, the JCM property for a compact space does not imply that \(X\) is separable. This result solves Problem 4.12 of the paper [\textit{A. V. Arhangel'skii} et al., Topology Appl. 169, 2--15 (2014; Zbl 1294.54005)] in the negative. Open questions are given in Section 4 of the paper.