Definition:Product Space (Topology)

Definition
Let $\mathbb S = \family {\struct {S_i, \tau_i} }_{i \mathop \in I}$ be an indexed family of topological spaces where $I$ is an arbitrary index set.

Let $X$ be the cartesian product of $\mathbb S$:
 * $\displaystyle S := \prod_{i \mathop \in I} S_i$

Let $\mathcal T$ be the Tychonoff topology on $S$.

The topological space $\struct {S, \mathcal T}$ is called the direct product of $\mathbb S$.

Also see

 * Product Topology is Topology


 * Definition:Initial Topology: $\mathcal T$, as defined here, is exactly the initial topology on $S$ with respect to $\family {\pr_i}_{i \mathop \in I}$, the indexed family of all projections $\pr_i: S \to S_i$.