Equicontinuous family of functions

Definition

Let $F$ be a family of functions from a topological space $X$ to a metric space $Y$. Then we say that the functions are equicontinuous if given any $x\in X$, and any $\epsilon \in \mathbb {R} ^{+}$, there is an open set $U\ni x$ such that $d(f(x),f(x'))<\epsilon \forall x\ \in U,f\in F$.