# Definition:Product Space (Topology)

## Definition

Let $T_1 = \left({S_1, \tau_1}\right)$ and $T_2 = \left({S_2, \tau_2}\right)$ be topological spaces.

Let $S_1 \times S_2$ be the cartesian product of $S_1$ and $S_2$.

The **product topology** $\tau$ for $S_1 \times S_2$ is the topology with basis $\mathcal B = \left\{{U_1 \times U_2: U_1 \in \tau_1, U_2 \in \tau_2}\right\}$.

The **product topology** $\tau$ is the same as the box topology for $S_1 \times S_2$.

It is also the same as the Tychonoff topology for $S_1 \times S_2$, which follows from Box Topology on Finite Product Space is Tychonoff Topology.

### General 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.

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

- Results about
**product spaces**can be found here.

## Sources

- 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $3.5$: Products: Definition $3.5.1$