Abstract:It is shown that if $C_p(X)$ admits a closure-preserving cover by closed $\sigma$-compact sets then $X$ is finite. If $X$ is compact and $C_p(X)$ has a closure-preserving cover by separable subspaces then $X$ is metrizable. We also prove that if $C_p(X,[0,1])$ has a closure-preserving cover by compact sets, then $X$ is discrete.
Keywords: closure-preserving covers, function spaces, compact spaces, pointwise convergence topology, topological game, winning strategy
AMS Subject Classification: 54C35