Uniformly continuous map

Definition

Between uniform spaces

Suppose $(X,\mathcal{U})$ and $(Y,\mathcal{V})$ are uniform spaces (in other words, $X$ and $Y$ are sets and $\mathcal{U}$ and $\mathcal{V}$ are uniform structures on $X$ and $Y$ respectively). A function $f:X\to Y$ is termed a uniformly continuous map if the following holds: For any $V\in \mathcal{V}$ (i.e., for every entourage of $Y$) there exists a $U\in \mathcal{U}$ such that $(a,b)\in U⟹(f(a),f(b))\in V$.

Between metric spaces

Suppose $(X,d_X)$ and $(Y,d_Y)$ are metric spaces (in other words, $X$ and $Y$ are sets and $d_X$ and $d_Y$ are metrics on $X$ and $Y$ respectively). A function $f:X\to Y$ is termed a uniformly continuous map if the following holds:

$\mathrm{\forall }\epsilon >0\mathrm{\exists }\delta >0:|d_X(a,b)|<\delta ⟹|d_Y(f(a),f(b))|<\epsilon$.