Corollary 7.3.3.label Let $X$ be a topological space, $\sigma \subset 2^{X}$ be an ideal such that $X$ is $\sigma$-generated, and $Y$ be a uniform space, then $C(X; Y) \subset Y^{X}$ is closed with respect to the $\sigma$-uniformity.
Proof. Let $f \in \overline{C(X; Y)}\subset Y^{X}$ with respect to the $\sigma$-uniformity. By Proposition 7.3.2, $f \in C(S; Y)$ for all $S \in \sigma$, so $f \in C(X; Y)$ by (3) of Definition 5.1.8.$\square$