\begin{definition}[Definition:Product Topology/Finite Product]
Let $I$ be a finite indexing set.
Let $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$ be an indexed family of topological spaces indexed by $I$.
Let $\XX$ be the cartesian product of $\family {X_i}_{i \mathop \in I}$:
:$\ds \XX := \prod_{i \mathop \in I} X_i$
For each $i \in I$, let $\pr_i: \XX \to X_i$ denote the $i$th projection on $\XX$:
:$\forall \family {x_j}_{j \mathop \in I} \in \XX: \map {\pr_i} {\family {x_j}_{j \mathop \in I} } = x_i$
The '''product topology''' on $\XX$ is defined as the initial topology $\tau$ on $\XX$ with respect to $\family {\pr_i}_{i \mathop \in I}$.
That is, $\tau$ is the topology generated by:
:$\SS = \set {\pr_i^{-1} \sqbrk U: i \in I, U \in \tau_i}$
where $\pr_i^{-1} \sqbrk U$ denotes the preimage of $U$ under $\pr_i$.
\end{definition}