product topology preserves the Hausdorff property

TheoremMathworldPlanetmath Suppose {Xα}αA is a collectionMathworldPlanetmath of Hausdorff spaces. Then the generalized Cartesian product αAXα equipped with the product topology is a Hausdorff space.

Proof. Let Y=αAXα, and let x,y be distinct points in Y. Then there is an index βA such that x(β) and y(β) are distinct points in the Hausdorff space Xβ. It follows that there are open sets U and V in Xβ such that x(β)U, y(β)V, and UV=. Let πβ be the projection operator YXβ defined here ( By the definition of the product topology, πβ is continuousPlanetmathPlanetmath, so πβ-1(U) and πβ-1(V) are open sets in Y. Also, since the preimageMathworldPlanetmath commutes with set operationsMathworldPlanetmath (, we have that

πβ-1(U)πβ-1(V) = πβ-1(UV)
= .

Finally, since x(β)U, i.e., πβ(x)U, it follows that xπβ-1(U). Similarly, yπβ-1(V). We have shown that U and V are open disjoint neighborhoodsMathworldPlanetmathPlanetmath of x respectively y. In other words, Y is a Hausdorff space.

Title product topology preserves the Hausdorff property
Canonical name ProductTopologyPreservesTheHausdorffProperty
Date of creation 2013-03-22 13:39:40
Last modified on 2013-03-22 13:39:40
Owner archibal (4430)
Last modified by archibal (4430)
Numerical id 7
Author archibal (4430)
Entry type Theorem
Classification msc 54B10
Classification msc 54D10