# Definition:Product Topology

## Definition

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

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

- $\ds \XX := \prod_{i \mathop \in I} X_i$

For each $i \in I$, let $\pr_i: \XX \to X_i$ denote the $i$th projection on $\XX$:

- $\forall \family {x_j}_{j \mathop \in I} \in \XX: \map {\pr_i} {\family {x_j}_{j \mathop \in I} } = x_i$

The **product topology** on $\XX$ is defined as the initial topology $\tau$ on $\XX$ with respect to $\family {\pr_i}_{i \mathop \in I}$.

That is, $\tau$ is the topology generated by:

- $\SS = \set {\pr_i^{-1} \sqbrk U: i \in I, U \in \tau_i}$

where $\pr_i^{-1} \sqbrk U$ denotes the preimage of $U$ under $\pr_i$.

### Two 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 $S_2$.

The **product topology** $\tau$ on $S_1 \times S_2$ is the topology generated by the natural basis:

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

### Finite Product

Let $I$ be a finite indexing set.

Let $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$ be an indexed family of topological spaces indexed by $I$.

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

- $\ds \XX := \prod_{i \mathop \in I} X_i$

For each $i \in I$, let $\pr_i: \XX \to X_i$ denote the $i$th projection on $\XX$:

- $\forall \family {x_j}_{j \mathop \in I} \in \XX: \map {\pr_i} {\family {x_j}_{j \mathop \in I} } = x_i$

The **product topology** on $\XX$ is defined as the initial topology $\tau$ on $\XX$ with respect to $\family {\pr_i}_{i \mathop \in I}$.

That is, $\tau$ is the topology generated by:

- $\SS = \set {\pr_i^{-1} \sqbrk U: i \in I, U \in \tau_i}$

where $\pr_i^{-1} \sqbrk U$ denotes the preimage of $U$ under $\pr_i$.

### Natural Sub-Basis

By definition of the initial topology on $\XX$ with respect to $\family {\pr_i}_{i \mathop \in I}$, $\tau$ is generated by the natural sub-basis.

The **natural sub-basis on $\XX$** is defined as:

- $\SS = \set {\pr_i^{-1} \sqbrk U: i \in I, \, U \in \tau_i}$

### Natural Basis

Let $\SS$ be the natural sub-basis on $X$:

- $\SS = \set {\pr_i^{-1} \sqbrk U: i \in I, \, U \in \tau_i}$

The **natural basis on $\SS$** is defined as the basis generated by $\SS$.

### 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**.

## 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

- Definition:Product Space (Topology): the topological space $\struct {X, \tau}$

- Natural Basis of Product Topology: $\tau$ is generated from the basis $\BB$ of cartesian products of the form $\ds \prod_{i \mathop \in I} U_i$ where:

- for all $i \in I : U_i \in \tau_i$
- for all but finitely many indices $i$: $U_i = X_i$

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

- Results about
**the product topology**can be found here.

### Relation between Product and Box Topology

- Results about the relation between the
**product topology**and the**box topology**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