Uniformly Equicontinuous Family of Maps Between Metric Spaces

Let $(X,d_{X})$ and $(Y,d_{Y})$ be metric spaces and let $\mathcal{F}$ be a set of maps from $X$ to $Y$.

The family $\mathcal{F}$ is uniformly equicontinuous if for every real number $\varepsilon>0$ there is a real number $\delta>0$ such that

$$d_{Y}\bigl(f(x),f(x')\bigr)<\varepsilon$$

for every $f\in\mathcal{F}$ and all $x,x'\in X$ satisfying $d_{X}(x,x')<\delta$.