Product topology explained

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 $\family {\pr_i}_{i \mathop \in I}$:

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

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 $\family {\pr_i}_{i \mathop \in I}$:

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

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 producttopological productTychonoff 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 Steenand J. Arthur Seebach, Jr.: Counterexamples in Topology[2nd ed.]... [previous]... [next]: Part $\text I$: Basic Definitions: Section $1$: General Introduction: Functions

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