Definition 6.5.1 (Equicontinuous).label Let $X$ be a topological space, $(Y, \fU)$ be a uniform space, $\cf \subset Y^{X}$, and $x \in X$, then $\cf$ is equicontinuous at $x$ if for every $U \in \fU$, there exists $V \in \cn_{X}(x)$ such that $(f(x), f(y)) \in U$ for all $y \in V$ and $f \in \cf$.

The set $\cf \subset C(X; Y)$ is equicontinuous if it is equicontinuous at every point in $x$.