Proposition 7

Let X be a topological space, let $\mathfrak{S}$ be a set of subsets of X and let Y be a uniform space. If Y is Hausdorff and if the union B of the sets of $\mathfrak{S}$ is dense in X, then $C_{\mathfrak{S}}(X; Y)$ is Hausdorff.

For if $(u, v)$ belongs to all the sets $W(A, V)$, where $A \in \mathfrak{S}$ and V is an entourage of Y, the hypothesis that Y is Hausdorff tells us that $u(x) = v(x)$ for all $x \in B$; if $u$ and $v$ are continuous, then $u = v$ by the principle of extension of identities (Chapter I, § 8, no. 1, Proposition 2, Corollary 1).

In particular, on $C(X; Y)$, the topology of pointwise convergence in a dense subset of X is Hausdorff.