Proposition 5.6.5. Let $X$ be a set, $\seqi{Y}$ be uniform spaces, $\seqi{f}$ with $f_{i}: X \to Y_{i}$ for each $i \in I$. Let $\fU$ be the initial uniformity on $X$ induced by $\seqi{f}$, and $(\wh X, \iota_{X})$ be the Hausdorff completion of $X$.
For each $i \in I$, let $(\wh Y_{i}, \iota_{i})$ be the Hausdorff completion of $Y_{i}$, then there exists a unique $F_{i} \in UC(\wh X; \wh Y_{i})$ such that the following diagram commutes
\[\xymatrix{ \wh X \ar@{->}[r]^{F_i} & \wh Y \\ X \ar@{->}[r]_{f_i} \ar@{->}[u] & Y \ar@{->}[u] }\]The uniformity of $\wh X$ is the initial uniformity induced by $\seqi{F}$.
There exists a unique $F \in UC(X; \prod_{i \in I}\wh Y_{i})$ and $\ol{F}\in UC(\wh X; \prod_{i \in I}\wh Y_{i})$ such that the following diagram commutes
\[\xymatrix{ & \prod_{i \in I} \wh Y_i \ar@{->}[rd]^{\pi_i} & \\ & \wh X \ar@{->}[r]^{F_i} \ar@{->}[u]_{\overline{F}} & \wh Y_i \\ X \ar@{->}[ruu]^{F} \ar@{->}[rr]_{f_i} \ar@{->}[ru]_{\iota_X} & & Y_i \ar@{->}[u]_{\iota_i} }\]Moreover, $\ol{F}(\wh X) = \overline{F(X)}$, and $\ol{F}$ is an embedding.
In particular, by Proposition 4.5.6, there is a natural isomorphism
induced by extending the identity on $\pi_{i \in I}X_{i}$.
Proof. (1): By (U) of Definition 5.6.1.
(2), (3): By (U) of the product, there exists $f \in UC(X; \prod_{i \in I}Y_{i})$ such that the following diagram commutes:
First suppose that $X$ and $\seqi{Y}$ are Hausdorff. For each $i \in I$, $\iota_{i} \circ \pi_{i} \in UC(\prod_{i \in I}Y_{i}; \wh Y_{i})$, so by (U) of Proposition 5.5.2, there exists a unique $\iota_{P} \in UC(\prod_{i \in I}Y_{i}; \prod_{i \in I}\wh Y_{i})$ such that the following diagram commutes
for all $i \in I$. By Lemma 5.6.2, each $\iota_{i} \in UC(Y_{i}; \wh Y_{i})$ is an embedding, so Proposition 4.7.4 implies that $\iota_{P}$ is an embedding as well.
Since $X$ has the initial topology, $f: X \to \prod_{i \in I}Y_{i}$ is an embedding by Proposition 4.7.3. Thus the composition $\iota_{P} \circ f$ is an embedding. As $\prod_{i \in I}\wh Y_{i}$ is complete by Proposition 5.5.2 and Proposition 4.8.3, $\overline{\iota_P \circ f(X)}\subset \prod_{i \in I}\wh Y_{i}$ is a complete Hausdorff uniform space. Let $Z$ be a complete Hausdorff uniform space and $g \in UC(X; Z)$, then Theorem 5.5.6 implies that there exists a unique $G \in UC(\overline{\iota_P \circ f(X)}; Z)$ such that the following diagram commutes
Thus $(\overline{\iota_P \circ f(X)}, \iota \circ f)$ satisfies (1), (2), and (U) of the Hausdorff completion. Therefore $\wh X$ may be identified as a subspace of $\prod_{i \in I}\wh Y_{i}$ as follows:
In which case, $\wh X$ must be equipped with the initial topology induced by the projection maps.
Now assume that $X$ and $Y$ are arbitrary. Let $X'$ and $\seqi{Y'}$ be the Hausdorff spaces associated with $X$ and $\seqi{Y}$, respectively. By Proposition 5.6.4, there exists $\seqi{f'}$ such that the following diagram commutes
for all $i \in I$. By (5) of Definition 5.6.1, there is a correspondence between the uniformities of $X$ and $X'$, and $Y$ and $Y'$. Thus $X'$ is equipped with the initial uniformity genereated by $\seqi{f'}$.$\square$