Proposition 9

Let $X$ be a topological space, $Y$ a uniform space. Then the mapping $(f, x) \to f(x)$ of $\mathscr{C}_u(X; Y) \times X$ into $Y$ is continuous.

Let $f_0 : X \to Y$ be a continuous mapping, let $x_0$ be a point of $X$ and let $V$ be an entourage of $Y$. The set $T$ of continuous mappings $f : X \to Y$ such that $(f(x), f(x_0)) \in V$ for all $x \in X$ is a neighbourhood of $f_0$ in $\mathscr{C}_u(X; Y)$. On the other hand, since $f_0$ is continuous, there is a neighbourhood $U$ of $x_0$ in $X$ such that $(f_0(x), f_0(x_0)) \in V$ for all $x \in U$. Consequently we have $(f(x), f(x_0)) \in V$ whenever $(f, x) \in T \times U$, and the result is proved.