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单词 InjectiveImagesOfBaireSpace
释义

injective images of Baire space


Every uncountable Polish spaceMathworldPlanetmath is, up to a countableMathworldPlanetmath subset, an injective image of Baire space 𝒩.

Theorem.

Let X be an uncountable Polish space. Then, there is a one-to-one and continuous functionMathworldPlanetmathPlanetmath f:NX such that Xf(N) is countable.

Although the inversePlanetmathPlanetmathPlanetmath f-1:f(𝒩)𝒩 will not generally be continuous, it is at least Borel measurable. It can be shown that this is true for all one-to-one and continuous functions between Polish spaces, although here it follows directly from the construction of f (http://planetmath.org/ProofOfInjectiveImagesOfBaireSpace).

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更新时间:2025/5/4 16:01:13