Corollary 3

Let $X$ be a topological space which is either metrizable or locally compact, and let $Y$ be a uniform space. Then $C(X; Y)$ is closed in the uniform space $F_c(X; Y)$; if in addition $Y$ is complete, the uniform space $C_c(X; Y)$ is complete.

By virtue of Corollary 2 it is enough to show that, if we take $\mathscr{S}$ to be the set of compact subsets of $X$, we have $\tilde{C}_\mathscr{S}(X; Y) = C(X; Y)$ in both cases under consideration. This is clear if $X$ is locally compact. If $X$ is metrizable, and $u : X \to Y$ is a mapping whose restriction to every compact subset of $X$ is continuous, then for each $x \in X$ and each sequence $(z_n)$ of points of $X$ which converges to $x$, we have $u(x) = \lim_{n \to \infty} u(z_n)$, and therefore $u$ is continuous at $x$ (Chapter IX, § 2, no. 6, Proposition 10).

Note that the argument above applies whenever every point of $x$ has a countable fundamental system of neighbourhoods.