# Definition:Product Space (Topology)

## 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 $\family {S_i}_{i \mathop \in I}$:

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

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

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 $\struct {X_i, \tau_i}$ are called the **factors** of $\struct {\XX, \tau}$, and can be referred to as **factor spaces**.

### $2$ Factor Spaces

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 product topology on $S_1 \times S_2$.

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

## Also known as

The **product topology** is also known as the **Tychonoff topology**, named for Andrey Nikolayevich Tychonoff.

While both of these terms are commonly used on $\mathsf{Pr} \infty \mathsf{fWiki}$, the preference is for **product topology**.

Various other terms can be found in the literature for the **product space**, for example:

**direct product****topological product****Tychonoff product**

but these terms are less precise, and there exists the danger of confusion with other similar uses of these terms in different contexts.

Note that the **topological space** $\struct {\XX, \tau}$ itself is never referred to as a **Tychonoff space**.

This is because a **Tychonoff space** is a different concept altogether.

## Also see

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

## Sources

- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction: Functions - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**Cartesian product** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**Cartesian product**