Definition 1

The uniformity on the set $\mathscr{F}(X; Y)$ which has as a fundamental system of entourages the set of subsets $W(V)$, where $V$ runs through the set of entourages of $Y$, is called the uniformity of uniform convergence. The topology it induces is called the topology of uniform convergence. If a filter $\Phi$ on $\mathscr{F}(X; Y)$ converges to an element $u_0$ with respect to this topology, $\Phi$ is said to converge uniformly to $u_0$.

Note that the topology of uniform convergence on $\mathscr{F}(X; Y)$ depends on the uniform structure of $Y$ and not merely on the topology of $Y$ (Exercise 4).

The uniform space obtained by endowing $\mathscr{F}(X; Y)$ with the uniformity of uniform convergence is denoted by $\mathscr{F}_u(X; Y)$.