product topology

Definition

Let ${((X_{\alpha },{𝒯}_{\alpha }))}_{\alpha \in A}$be a family of topological spaces, and let $Y$ be theCartesian product (http://planetmath.org/GeneralizedCartesianProduct)of the sets $X_{\alpha }$, that is

$$Y=\prod _{\alpha \in A}{X}_{\alpha }.$$

Recall that an element $y\in Y$ is a function$y:A\to {\bigcup }_{\alpha \in A}{X}_{\alpha }$such that $y(\alpha )\in X_{\alpha }$ for each $\alpha \in A$,and that for each $\alpha \in A$the projection map ${\pi }_{\alpha }:Y\to X_{\alpha }$is defined by ${\pi }_{\alpha }(y)=y(\alpha )$ for each $y\in Y$.

The (Tychonoff) product topology $𝒯$ for $Y$is defined to be the initial topology with respect to the projection maps;that is,$𝒯$ is the smallest topology such that each ${\pi }_{\alpha }$ is continuous (http://planetmath.org/Continuous).

Subbase

If $U\subseteq X_{\alpha }$ is open,then ${\pi }_{\alpha }^{-1}(U)$ is an open set in $Y$.Note that this is the set of all elements of $Y$in which the $\alpha$ component is restricted to $U$and all other components are unrestricted.The open sets of $Y$ are the unions of finite intersections of such sets.That is,

$$\{{\pi }_{\alpha }^{-1}(U)\mid \alpha \in A\text{ and }U\in {𝒯}_{\alpha }\}$$

is a subbase for the topology on $Y$.

Theorems

The following theorems assume the product topology on${\prod }_{\alpha \in A}{X}_{\alpha }$.Notation is as above.

Theorem 1

Let $Z$ be a topological spaceand let $f\mathrm{:}Z\mathrm{\to }{\mathrm{\prod }}_{\alpha \mathrm{\in }A}{X}_{\alpha }$ be a function.Then $f$ is continuous if and only if ${\pi }_{\alpha }\mathrm{\circ }f$ is continuousfor each $\alpha \mathrm{\in }A$.

Theorem 2

The product topology on ${\mathrm{\prod }}_{\alpha \mathrm{\in }A}{X}_{\alpha }$is the topology induced by the subbase

$$\{{\pi }_{\alpha }^{-1}(U)\mid \alpha \in A\mathit{\text{ and }}U\in {𝒯}_{\alpha }\}.$$

Theorem 3

The product topology on ${\mathrm{\prod }}_{\alpha \mathrm{\in }A}{X}_{\alpha }$is the topology induced by the base

$$\left\{\bigcap _{\alpha \in F}{\pi }_{\alpha }^{-1}(U_{\alpha })\right|F\mathit{\text{ is a finite subset of }}A\mathit{\text{ and }}{U}_{\alpha }\in {𝒯}_{\alpha }\mathit{\text{ for each }}\alpha \in F\}.$$

Theorem 4

A net ${\mathrm{(}{x}_i\mathrm{)}}_{i\mathrm{\in }I}$ in ${\mathrm{\prod }}_{\alpha \mathrm{\in }A}{X}_{\alpha }$ converges to $x$ if and only if each coordinate ${\mathrm{(}{x}_i^{\alpha }\mathrm{)}}_{i\mathrm{\in }I}$ converges to $x^{\alpha }$ in $X_{\alpha }$.

Theorem 5

Each projection map ${\pi }_{\alpha }\mathrm{:}{\mathrm{\prod }}_{\alpha \mathrm{\in }A}{X}_{\alpha }\mathrm{\to }{X}_{\alpha }$is continuous and open (http://planetmath.org/OpenMapping).

Theorem 6

For each $\alpha \mathrm{\in }A$, let $A_{\alpha }\mathrm{\subseteq }{X}_{\alpha }$.Then

$$\overline{\prod _{\alpha \in A}{A}_{\alpha }}=\prod _{\alpha \in A}\overline{A_{\alpha }}.$$

In particular, any product of closed sets is closed.

Theorem 7

(Tychonoff’s Theorem)If each $X_{\alpha }$ is compact, then ${\mathrm{\prod }}_{\alpha \mathrm{\in }A}{X}_{\alpha }$ is compact.

Comparison with box topology

There is another well-known way to topologize $Y$, namely the box topology.The product topology is a subset of the box topology;if $A$ is finite, then the two topologies are the same.

The product topology is generally more useful than the box topology.The main reason for this can be expressed in terms of category theory:the product topology is the topology of thedirect categorical product (http://planetmath.org/CategoricalDirectProduct)in the category Top (see Theorem 1 above).