Uniformly continuous map

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
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:

$$\forall \ \varepsilon > 0 \ \exists \delta > 0 : d_X(a,b) < \delta \implies d_Y(f(a),f(b)) < \varepsilon$$.