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单词 ProofForOneEquivalentStatementOfBaireCategoryTheorem
释义

proof for one equivalent statement of Baire category theorem


First, let’s assume Baire’s category theoremMathworldPlanetmath and prove the alternative statement.
We have B=n=1Bn, with int(Bk¯)=k𝐍.
Then

X=X-int(Bk¯)=X-Bk¯¯k𝐍

Then X-Bk¯ is dense in X for every k. Besides, X-Bk¯ is open because X is open andBk¯ closed. So, by Baire’s Category Theorem, we have that

n=1(X-Bn¯)=X-n=1Bn¯

is dense in X.But Bn=1Bn¯X-n=1Bn¯X-B, and thenX=X-n=1Bn¯¯X-B¯=X-int(B)int(B)=.

Now, let’s assume our alternative statement as the hypothesisMathworldPlanetmathPlanetmath, and let (Bk)kN be a collectionMathworldPlanetmath of open dense sets ina complete metric space X.Then int(X-Bk¯)=int(X-int(Bk))=int(X-Bk)=X-Bk¯= and soX-Bk is nowhere dense for every k.

Then X-n=1Bn¯=int(X-n=1Bn)=int(n=1X-Bn)= due to our hypothesis. Hence Baire’s category theorem holds.
QED

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更新时间:2025/5/5 5:20:14