Definition:Product Space (Topology)

Definition
Let $\mathbb X = \left \langle {\left({X_i, \tau_i}\right)}\right \rangle_{i \in I}$ be a family of topological spaces where $I$ is an arbitrary index set.

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

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

The topological space $\left({X, \mathcal T}\right)$ is called the direct product of $\mathbb X$.

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

Also see

 * Product Topology is Topology