Definition:Product Space (Topology)

Definition
Let $\mathbb S = \left \langle {\left({S_i, \tau_i}\right)}\right \rangle_{i \in I}$ be a 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 $\left({S, \mathcal T}\right)$ is called the direct product of $\mathbb S$.

Factor Space
Each of the topological spaces $\left({S_i, \tau_i}\right)$ are called the factors of $\left({S, \mathcal T}\right)$, and can be referred to as factor spaces.

Also see

 * Product Topology is Topology


 * Definition:Initial Topology: $\mathcal T$, as defined here, is exactly the initial topology on $S$ with respect to $\left \langle {\operatorname{pr}_i}\right \rangle_{i \in I}$, the family of all projections $\operatorname{pr}_i: S \to S_i$.