Definition 5.6.3 (Associated Hausdorff Uniform Space, [Proposition 2.8.16, Bou13]). Let $(X, \fU)$ be a uniform space, then there exists $(X', i)$ such that:
$X'$ is a Hausdorff uniform space.
$i \in UC(X; X')$.
For any pair $(Y, f)$ satisfying (1) and (2), there exists a unique $f' \in UC(X'; Y)$ such that the following diagram commutes
\[\xymatrix{ X' \ar@{->}[rd]^{F} & \\ X \ar@{->}[r]_{f} \ar@{->}[u]^{i} & Y }\]
known as the Hausdorff uniform space associated with $(X, \fU)$.
Proof. Let $(\wh X, \iota)$ be the Hausdorff completion of $X$, $X' = \iota(X)$, and $i = \iota$, then $(X', i)$ satisfies (1) and (2).
(U): Let $(\wh Y, \iota)$ be the Hausdorff completion of $Y$. Using Lemma 5.6.2, identify $Y$ as a subspace of $\wh Y$. By (U) of the Hausdorff completion, there exists a unique $\ol F \in UC(\wh X; \wh Y)$ such that the following diagram commutes:
Since $\ol F(X') \subset \iota(Y) = Y'$, $F = \ol F|_{X'}\in UC(X'; Y')$ is continuous. By Lemma 5.6.2, $\iota \in UC(Y; Y')$ is a homeomorphism. Upon identifying $Y$ with $Y'$, $F \in UC(X'; Y)$ is the desired map.$\square$