Proposition 1

If $Y$ is Hausdorff and $\mathscr{S}$ is a covering of $X$, then the space $\mathscr{F}_\mathscr{S}(X; Y)$ is Hausdorff.

Let $u, v$ be two elements of $\mathscr{F}_\mathscr{S}(X; Y)$ such that $(u, v) \in W(A, V)$ for every entourage $V$ of $Y$ and every $A \in \mathscr{S}$. Since $Y$ is Hausdorff it follows that $u$ and $v$ coincide on every set $A \in \mathscr{S}$, and since $\mathscr{S}$ covers $X$ we must have $u = v$.