Definition 5.2.1 (Uniform Continuity). Let $(X, \fU)$ and $(Y, \mathfrak{V})$ be uniform spaces and $f: X \to Y$, then the following are equivalent:
For every $V \in \mathfrak{V}$, there exists $V' \in \fU$ such that $(f(x), f(y)) \in V$ whenever $(x, y) \in V'$.
For every $V \in \mathfrak{V}$, $(f \times f)^{-1}(V) \in \fU$.
If the above holds, then $f$ is a uniformly continuous function.
The collection $UC(X; Y)$ denotes the set of all uniformly continuous functions from $X$ to $Y$.
Proof. $(1) \Rightarrow (2)$: Since $(f(x), f(y)) \in V$ whenever $(x, y) \in V'$, $V' \subset f^{-1}(V)$. By (F1) of $\fU$, $f^{-1}(V) \in \fU$.
$(2) \Rightarrow (1)$: Take $V' = (f \times f)^{-1}(V)$, then $(f(x), f(y)) \in V$ whenever $(x, y) \in V'$.$\square$