Remark 1

Likewise, the topology induced on $C(X; Y)$ by the product topology on $Y^X = \mathscr{F}(X; Y)$ is called the topology of pointwise convergence ($Y$ being not necessarily uniformizable); it is generated by sets of the form $T(\{x\}, U)$ as $x$ runs through $X$ and $U$ runs through the set of all open subsets of $Y$, and it is therefore coarser than the compact-open topology. We deduce that, if $Y$ is Hausdorff, the space $C_c(X; Y)$ is Hausdorff (Chapter I, § 8, no. 1, Proposition 5, Corollary).