Function spaces jointly metrizable on compacta

Abstract

If Cp(X) is jointly metrizable on compacta, then p(X)≤ω but ω1 need not be a caliber of X. If X is either submetrizable or a P-space, then Cp(Cp(X)) is jointly metrizable on compacta and, in particular, all compact subsets of Cp(Cp(X)) are metrizable. We show that for any dyadic compact X, the space Cp(X) is jointly metrizable on compacta. Therefore, the JCM property of Cp(X) for a compact space X does not imply that X is separable. If X is a compact space of countable tightness and Cp(X) is jointly metrizable on compacta, then it is independent of ZFC whether X must be separable.