请输入您要查询的字词:

 

单词 ProofOfPolishSpacesUpToBorelIsomorphism
释义

proof of Polish spaces up to Borel isomorphism


We show that every uncountable Polish spaceMathworldPlanetmath X is Borel isomorphic to the real numbers.First, there exists a continuousMathworldPlanetmathPlanetmath one-to-one and injective function f from Baire spacePlanetmathPlanetmath 𝒩 to X such that Xf(𝒩) is countableMathworldPlanetmath, and such that the inversePlanetmathPlanetmathPlanetmath from f(𝒩) to 𝒩 is Borel measurable (see here (http://planetmath.org/InjectiveImagesOfBaireSpace)).Letting S be any countably infiniteMathworldPlanetmath subset of X, the same result can be applied to XS, which is also a Polish space.So, there is a continuous and one-to-one function f:𝒩XS such that SXf(𝒩) is countable and such that the inverse defined on XS is Borel.Then, S contains S and is countably infinite.Hence, there is a invertible function g from ={1,2,} to S. Under the discrete topology on this is necessarily a continuous function with Borel measurable inverse. By combining the functions f and g, this gives a continuous, one-to-one and onto function from the disjoint unionMathworldPlanetmath (http://planetmath.org/TopologicalSum)

u:𝒩X

with Borel measurable inverse.Similarly, the set of real numbers with the standard topology is an uncountable Polish space and, therefore, there is a continuous function v from 𝒩 to with Borel inverse. So, vu-1 gives the desired Borel isomorphism from X to .

随便看

 

数学辞典收录了18232条数学词条,基本涵盖了常用数学知识及数学英语单词词组的翻译及用法,是数学学习的有利工具。

 

Copyright © 2000-2023 Newdu.com.com All Rights Reserved
更新时间:2025/5/4 8:34:11