Corollary 2

Let $X$ be a topological (resp. uniform) space, let $Y$ be a complete Hausdorff uniform space and let $H$ be an equicontinuous (resp. uniformly equicontinuous) subset of $C(X; Y)$. Suppose that $H(x)$ is relatively compact in $Y$ for all $x \in D$, where $D$ is a dense subset of $X$. Then $H$ is relatively compact in $C(X; Y)$ with respect to the topology of compact (resp. precompact) convergence.

It is enough to show that $H(x)$ is relatively compact for all $x \in X$, for we can then apply Corollary 1. Since $Y$ is complete it is enough to show that $H(x)$ is precompact for all $x \in X$. Now if $V$ is any symmetric entourage of $Y$, there is a neighbourhood $U$ of $x$ such that $(u(x), u(x')) \in V$ for all $x' \in U$ and all $u \in H$. By hypothesis there exists $x' \in U \cap D$, and since $H(x')$ is relatively compact in $Y$, there exists a finite number of points $y_k \in Y$ such that $H(x')$ is contained in the union of the sets $V(y_k)$; hence $H(x)$ is contained in the union of the sets $V(y_k)$, and the proof is complete.