Proposition 5.7.4.label Let $\seqi{X}$, $\seqi{Y}$ be topological/uniform spaces and $\seqi{f}$ such that $f_{i} \in C(X_{i}; Y_{i})$ is an embedding for all $i \in I$, then
- (1)
There exists a unique $f \in C(\prod_{i \in I}X_{i}; \prod_{i \in I}Y_{i})$/$f \in UC(\prod_{i \in I}X_{i}; \prod_{i \in I}Y_{i})$ such that the following diagram commutes
\[\xymatrix{ \prod_{i \in I}Y_i \ar@{->}[r]^{\pi_i} & Y_i \\ \prod_{i \in I} X_i \ar@{->}[r]_{\pi_i} \ar@{->}[u]^{f} & X_i \ar@{^{(}->}[u]_{f_i} }\]for all $i \in I$.
- (2)
$f$ is an embedding.
Proof. (1): By (U) of Definition 5.7.1/Definition 6.2.5.
(2): Consider the following diagram
\[\xymatrix{ \prod_{i \in I}Y_i \ar@{->}[rd]^{\pi_i} & \\ \iota_P(\prod_{i \in I}X_i) \ar@{->}[u] & Y_i \\ \prod_{i \in I} X_i \ar@{->}[r]_{\pi_i} \ar@{->}[u]^{f} & X_i \ar@{^{(}->}[u]_{f_i} }\]
Since each $X_{i} \to Y_{i}$ is an embedding, the composition
\[\xymatrix{ \iota_P(\prod_{i \in I}X_i) \ar@{->}[r]^{\pi_i} & Y_i \ar@{->}[r]^{f_i^{-1}} & X_i }\]
is continuous/uniformly continuous. By (U) of Definition 5.7.1/Definition 6.2.5, $f$ is an embedding.$\square$