Definition 2

Let X and Y be two uniform spaces. A subset H of $\mathscr{F}(X; Y)$ is said to be uniformly equicontinuous if, for each entourage V of Y, there exists an entourage U of X such that we have $(f(x), f(x')) \in V$ whenever $(x, x') \in U$ and $f \in H$.

A family $(f_i)_{i \in I}$ of mappings of X into Y is said to be equicontinuous at a point $x_0$ (resp. equicontinuous, uniformly equicontinuous) if the set of the $f_i$ is equicontinuous at $x_0$ (resp. equicontinuous, uniformly equicontinuous).

It is clear that if $H \subset \mathscr{F}(X; Y)$ is equicontinuous at $x_0$, then each $f \in H$ is continuous at $x_0$; if H is equicontinuous, then each $f \in H$ is continuous on X, i.e. $H \subset C(X; Y)$. Likewise, if H is uniformly equicontinuous (X being a uniform space), every $f \in H$ is uniformly continuous on X. It is also clear that a uniformly equicontinuous set of mappings is equicontinuous; but a set of uniformly continuous mappings can be equicontinuous without being uniformly equicontinuous (see Exercise 1; Corollary 2 to Proposition 1; and no. 2, Proposition 4).