Definition 4.7.1 (Product Topology). Let $\bracsn{(X_i, \topo_i)}_{i \in I}$ be a family of topological spaces, then the product topology $\topo$ is the initial topology generated by the projections $\seqi{\pi}$, and:
The family
\[\cb(\ce) = \bracs{\bigcap_{k = 1}^n \pi_{i_k}^{-1}(U_{k}) \bigg | U_{k} \in \topo_{i_k}, \seqf{i_k} \subset I, n \in \nat^+}\]is a base for $\topo$.
For any topological space space $Y$ and $\seqi{f}$ where $f_{i} \in C(Y; X_{i})$ for all $i \in I$, there exists a unique $f \in C(Y; X)$ such that the following diagram commutes
\[\xymatrix{ Y \ar@{->}[d]_{f} \ar@{->}[rd]^{f_i} & \\ X \ar@{->}[r]_{\pi_i} & X_i }\]for all $i \in I$.
Proof. (1): By (3) of Definition 4.1.7.
(U): Let $f \in \prod_{i \in I}f_{i}$, then $f$ is the unique function such that the diagrams commute. For each $\bigcap_{k = 1}^{n} \pi_{i_k}^{-1}(U_{k}) \in \topo$,
\[f^{-1}\paren{\bigcap_{k = 1}^n \pi_{i_k}^{-1}(U_k)}= \bigcap_{k = 1}^{n} f_{i_k}^{-1}(U_{k})\]
which is open in $Y$ by (O2).$\square$