Definition 1

Let $X$ be a topological space and $Y$ a uniform space. A subset $H$ of $\mathscr{F}(X; Y)$ is said to be equicontinuous at a point $x_0 \in X$ if, for each entourage $V$ of $Y$, there is a neighbourhood $U$ of $x_0$ in $X$ such that $(f(x_0), f(x)) \in V$ for all $x \in U$ and all $f \in H$. H is said to be equicontinuous if it is equicontinuous at every point of $X$.