uncountable Polish spaces contain Cantor space
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.
|Title||uncountable Polish spaces contain Cantor space|
|Date of creation||2013-03-22 18:48:33|
|Last modified on||2013-03-22 18:48:33|
|Last modified by||gel (22282)|