product topology


Let ((Xα,𝒯α))αA be a family of topological spacesMathworldPlanetmath, and let Y be the Cartesian product ( of the sets Xα, that is


Recall that an element yY is a function y:AαAXα such that y(α)Xα for each αA, and that for each αA the projection map πα:YXα is defined by πα(y)=y(α) for each yY.

The (TychonoffPlanetmathPlanetmath) 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 πα is continuousPlanetmathPlanetmath (


If UXα is open, then πα-1(U) is an open set in Y. Note that this is the set of all elements of Y in which the α component is restricted to U and all other components are unrestricted. The open sets of Y are the unions of finite intersectionsMathworldPlanetmathPlanetmath of such sets. That is,

{πα-1(U)αA and U𝒯α}

is a subbase for the topology on Y.


The following theorems assume the product topology on αAXα. Notation is as above.

Theorem 1

Let Z be a topological space and let f:ZαAXα be a function. Then f is continuous if and only if παf is continuous for each αA.

Theorem 2

The product topology on αAXα is the topology induced by the subbase

{πα-1(U)αA and U𝒯α}.
Theorem 3

The product topology on αAXα is the topology induced by the base

{αFπα-1(Uα)|F is a finite subset of A and Uα𝒯α for each αF}.
Theorem 4

A net (xi)iI in αAXα convergesPlanetmathPlanetmath to x if and only if each coordinate (xiα)iI converges to xα in Xα.

Theorem 5

Each projection map πα:αAXαXα is continuous and open (

Theorem 6

For each αA, let AαXα. Then


In particular, any productPlanetmathPlanetmathPlanetmath of closed setsPlanetmathPlanetmath is closed.

Theorem 7

(Tychonoff’s Theorem) If each Xα is compactPlanetmathPlanetmath, then αAXα 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 theoryMathworldPlanetmathPlanetmathPlanetmathPlanetmath: the product topology is the topology of the direct categorical product ( in the categoryMathworldPlanetmath Top (see Theorem 1 above).


  • 1 J. L. Kelley, General Topology, D. van Nostrand Company, Inc., 1955.
  • 2 J. Munkres, Topology (2nd edition), Prentice Hall, 1999.
Title product topology
Canonical name ProductTopology
Date of creation 2013-03-22 12:47:09
Last modified on 2013-03-22 12:47:09
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 38
Author CWoo (3771)
Entry type Definition
Classification msc 54B10
Synonym Tychonoff product topology
Related topic BoxTopology
Related topic GeneralizedCartesianProduct
Related topic ASpaceMathnormalXIsHausdorffIfAndOnlyIfDeltaXIsClosed
Related topic InitialTopology
Defines product