Proposition 6.5.5 ([Proposition 2.3.11, Bou13]).label Let $X$ be a uniform space, $Y$ be a complete Hausdorff uniform space, $A \subset X$ be a dense subset, and $f \in C(A; Y)$ be a continuous function, then the following are equivalent:
- (1)
There exists a continuous function $F \in C(X; Y)$ such that $F|_{A} = f$.
- (2)
$f$ is Cauchy continuous.
Proof. By Proposition 6.1.16, $Y$ is regular. Since $Y$ is complete, (2) is equivalent to (2) of Theorem 5.9.2. Therefore the proposition follows from Theorem 5.9.2.$\square$