1. The Baire category theorem - Aarhus Universitet

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Proof of the corollary Reminder ?: A subset E of S is dense when E ∩ U = ∅ for all open and non-empty subsets U. Examples: 1) Q is dense in R. The polynomials are dense in C[0,1]. Nullermændene er tætte i min ældste søns værelse. Proof of corollary using the theorem: If S = ∞ i=1 Ei, where each Ei is nowhere dense, observe that ∅ = ∞ i=1 Ec i ⊇ ∞ i=1 Ei c c and each Ei is dense and open. This contradicts the theorem. Klaus Thomsen 1. The Baire category theorem
Proof of Baire’s category theorem Let V1,V2,V3,... be open and dense subsets of S. We must show that ∞ i=1 Vi is dense in S when S is either a complete metric space or a locally compact Hausdorff space. Klaus Thomsen 1. The Baire category theorem
Page 1 and 2: 1. The Baire category theorem Klaus
Page 3 and 4: We read in W. Rudin: Functional Ana
Page 5 and 6: Definition of first and second cate
Page 11 and 12: A subset A of S is nowhere dense wh
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Page 21 and 22: Basic properties the notions of cat
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Page 35 and 36: Proof of the corollary Reminder ?:
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Page 47 and 48: On locally compact Hausdorff spaces
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1. The Baire category theorem - Aarhus Universitet

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