spaces homeomorphic to Baire space

Baire spacePlanetmathPlanetmath, 𝒩, is the set of all functionsMathworldPlanetmath x: together with the product topology. This is homeomorphic to the set of irrational numbers in the unit interval, with the homeomorphism f:𝒩(0,1) given by continued fractionDlmfMathworldPlanetmath expansion

Theorem 1.

Let I be an open intervalDlmfPlanetmath of the real numbers and S be a countableMathworldPlanetmath dense subset of I. Then, IS is homeomorphic to Baire space.

More generally, Baire space is uniquely characterized up to homeomorphism by the following properties.

Theorem 2.

A topological spaceMathworldPlanetmath X is homeomorphic to Baire space if and only if

  1. 1.

    It is a nonempty Polish spaceMathworldPlanetmath.

  2. 2.

    It is zero dimensional (

  3. 3.

    No nonempty open subsets are compactPlanetmathPlanetmath.

In particular, for an open interval I of the real numbers and countable dense subset SI, then IS is easily seen to satisfy these properties and Theorem 1 follows.

Title spaces homeomorphic to Baire space
Canonical name SpacesHomeomorphicToBaireSpace
Date of creation 2013-03-22 18:46:48
Last modified on 2013-03-22 18:46:48
Owner gel (22282)
Last modified by gel (22282)
Numerical id 7
Author gel (22282)
Entry type Theorem
Classification msc 54E50
Related topic PolishSpace
Related topic InjectiveImagesOfBaireSpace