Definition:Uniformly Equicontinuous

Definition
Let $X = \left({ A, d }\right)$ and $Y = \left({ B, \rho }\right)$ be metric spaces.

Let $\left\langle{ f_i }\right\rangle_{i \mathop \in I}$ be a family of  mappings $f_i: X \to Y$.

Then $\left\langle{ f_i }\right\rangle_{i \mathop \in I}$ is said to be uniformly equicontinuous on $S \subseteq A$ :
 * $\forall \epsilon \in \R_{>0} : \exists \delta \in \R_{>0} : \forall i \in I : \forall x, y \in A : \forall x, y \in S : d \left({ x, y }\right) < \delta \implies \rho \left({ f_i \left({ x }\right), f_i \left({ y }\right) }\right)< \epsilon$