Theorem 2

Let $X$ be a topological space, $Y$ a uniform space. Then the set $C(X; Y)$ of continuous mappings of $X$ into $Y$ is a closed subset of the space $F(X; Y)$ endowed with the topology of uniform convergence.

For each $x \in X$, the set of mappings of $X$ into $Y$ which are continuous at $x$ is closed in $F_u(X; Y)$ (Proposition 8, Corollary 1); hence the intersection $C(X; Y)$ of these sets is also closed.

This result may be expressed in the form that the uniform limit of continuous functions is continuous.