# Definition:Product Space (Topology)

## Definition

Let $\struct{S_1, \tau_1}$ and $\struct{S_2, \tau_2}$ be topological spaces.

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

Let $\tau$ be the Tychonoff topology on $S_1 \times S_2$.

From Natural Basis of Tychonoff Topology of Finite Product, $\tau$ is the topology generated by the basis

- $\BB = \set{U_1 \times U_2: U_1 \in \tau_1, U_2 \in \tau_2}$.

The topological space $\struct{S_1 \times S_2, \tau}$ is called the **product space** of $\struct{S_1, \tau_1}$ and $\struct{S_2, \tau_2}$.

## General Definition

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

Let $S$ be the cartesian product of $\family {S_i}_{i \mathop \in I}$:

- $\displaystyle S := \prod_{i \mathop \in I} S_i$

Let $\tau$ be the Tychonoff topology on $S$.

From Natural Basis of Tychonoff Topology, $\tau$ is generated from:

- the basis $\BB$ of cartesian products of the form $\displaystyle \prod_{i \mathop \in I} U_i$ where:
- for all $i \in I : U_i \in \tau_i$
- for all but finitely many indices $i : U_i = S_i$

The topological space $\struct{X, \tau}$ is called the **product space** of $\family {\struct {S_i, \tau_i} }_{i \mathop \in I}$.

## Factor Space

Each of the topological spaces $\struct{S_i, \tau_i}$ are called the **factors** of $\struct{S, \tau}$, and can be referred to as **factor spaces**.

## Also known as

The topology on the **product space** is also known as the **product topology** and both of the terms **product topology** and **Tychonoff topology** are commonly used on $\mathsf{Pr} \infty \mathsf{fWiki}$.

Other sources may use the terms **direct product**, **topological product** or **Tychonoff product** for the **product space**, but these terms are less precise and there exists the danger of confusion with other similar uses of these terms in different contexts.

## Also see

- Natural Basis of Tychonoff Topology
- Natural Basis of Tychonoff Topology of Finite Product
- Tychonoff Topology is Coarsest Topology such that Projections are Continuous
- Product Space is Product in Category of Topological Spaces

- 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 $3.5.1$