Exercise 7, § 3

Let X be a space which has a countable dense subset, let Y be a Hausdorff space in which every point has a countable fundamental system of neighbourhoods (resp. a metrizable space) and let H be a subset of $\mathscr{C}(X; Y)$. Show that if $\mathscr{T}$ is a topology on H which is finer than the topology of pointwise convergence in X and with respect to which H is compact, then every point of H has a countable fundamental system of neighbourhoods in the topology $\mathscr{T}$ (resp. $\mathscr{T}$ is metrizable). (Note that if D is any countable dense subset of X, then the topology $\mathscr{T}$ is finer than the topology of pointwise convergence in D and that the latter topology is Hausdorff.)