uncountable Polish spaces contain Cantor space

Cantor space is an example of a compactPlanetmathPlanetmath and uncountable Polish spaceMathworldPlanetmath. In fact, every uncountable Polish space contains Cantor space, as stated by the following theorem.


Let X be an uncountable Polish space. Then, it contains a subset S which is homeomorphic to Cantor space.

For example, the set of real numbers contains the Cantor middle thirds set (http://planetmath.org/CantorSet). Note that, being homeomorphic to Cantor space, S must be a compact and hence closed subset of X. The result is trivial in the case of Baire spacePlanetmathPlanetmath 𝒩, in which case we may take S to be the set of all s𝒩 satisfying sn{1,2} for all n. Then, for any uncountable Polish space X there exists a continuousPlanetmathPlanetmath and one-to-one function f:𝒩X (see here (http://planetmath.org/InjectiveImagesOfBaireSpace)). Then f gives a continuous bijection from S to f(S). The inverse function theorem (http://planetmath.org/InverseFunctionTheoremTopologicalSpaces) implies that f is a homeomorphism between S and f(S) and, therefore, f(S) is homeomorphic to Cantor space.

Title uncountable Polish spaces contain Cantor space
Canonical name UncountablePolishSpacesContainCantorSpace
Date of creation 2013-03-22 18:48:33
Last modified on 2013-03-22 18:48:33
Owner gel (22282)
Last modified by gel (22282)
Numerical id 6
Author gel (22282)
Entry type Theorem
Classification msc 54E50
Related topic PolishSpace