# Topology of uniform convergence

The topology on the space ${\mathcal F} ( X, Y)$ of mappings from a set $X$ into a uniform space $Y$ generated by the uniform structure on ${\mathcal F} ( X, Y)$, the base for the entourages of which are the collections of all pairs $( f, g) \in {\mathcal F} ( X, Y) \times {\mathcal F} ( X, Y)$ such that $( f ( x), g ( x)) \in v$ for all $x \in X$ and where $v$ runs through a base of entourages for $Y$. The convergence of a directed set $\{ f _ \alpha \} _ {\alpha \in A } \subset {\mathcal F} ( X, Y)$ to $f _ {0} \in {\mathcal F} ( X, Y)$ in this topology is called uniform convergence of $f _ \alpha$ to $f _ {0}$ on $X$. If $Y$ is complete, then ${\mathcal F} ( X, Y)$ is complete in the topology of uniform convergence. If $X$ is a topological space and ${\mathcal C} ( X, Y)$ is the set of all mappings from $X$ into $Y$ that are continuous, then ${\mathcal C} ( X, Y)$ is closed in ${\mathcal F} ( X, Y)$ in the topology of uniform convergence; in particular, the limit $f _ {0}$ of a uniformly-convergent sequence $f _ {n}$ of continuous mappings on $X$ is a continuous mapping on $X$.