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 $(a,b)\in U\implies (f(a),f(b))\in V$.

Between metric spaces

Further information: Uniformly continuous map of 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:

$\forall \ \varepsilon >0\ \exists \delta >0:d_{X}(a,b)<\delta \implies d_{Y}(f(a),f(b))<\varepsilon$.