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单词 UncountablePolishSpacesContainCantorSpace
释义

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.

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.

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更新时间:2025/5/4 12:44:48