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 $\family {S_i}_{i \mathop \in I}$:
 * $\displaystyle S := \prod_{i \mathop \in I} S_i$

Let $\tau$ be the Tychonoff topology on $S$.

From Natural Basis of Tychonoff Topology, $\tau$ is generated from:
 * the basis $\BB$ of cartesian products of the form $\displaystyle \prod_{i \mathop \in I} U_i$ where:
 * for all $i \in I : U_i \in \tau_i$
 * for all but finitely many indices $i : U_i = S_i$

The topological space $\struct{X, \tau}$ is called the product space of $\family {\struct {S_i, \tau_i} }_{i \mathop \in I}$.