\begin{definition}[Definition:Product Space (Topology)]
Let $\family {\struct {S_i, \tau_i} }_{i \mathop \in I}$ be an indexed family of topological spaces where $I$ is an arbitrary index set.
Let $S$ be the cartesian product of $\family {S_i}_{i \mathop \in I}$:
:$\ds S := \prod_{i \mathop \in I} S_i$
Let $\tau$ be the product topology on $S$.
The topological space $\struct {X, \tau}$ is called the '''product space''' of $\family {\struct {S_i, \tau_i} }_{i \mathop \in I}$.
\end{definition}