Definition:Product Topology/Factor Space

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

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

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.