Definition:Product Space (Topology)

Definition
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 $\family {\struct {S_i, \tau_i} }_{i \mathop \in I}$.