uncountable Polish spaces contain Cantor space
Cantor space is an example of a compact and uncountable Polish space
. In fact, every uncountable Polish space contains Cantor space, as stated by the following theorem.
Theorem.
Let be an uncountable Polish space. Then, it contains a subset 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, must be a compact and hence closed subset of .The result is trivial in the case of Baire space , in which case we may take to be the set of all satisfying for all .Then, for any uncountable Polish space there exists a continuous
and one-to-one function (see here (http://planetmath.org/InjectiveImagesOfBaireSpace)). Then gives a continuous bijection from to . The inverse function theorem (http://planetmath.org/InverseFunctionTheoremTopologicalSpaces) implies that is a homeomorphism between and and, therefore, is homeomorphic to Cantor space.