1. The Baire category theorem - Aarhus Universitet

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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. Consider an open non-empty subset B0 of S. We want to show that ∞ i=1 Vi ∩ B0 = ∅. Since V1 is dense in S, B0 ∩ V1 = ∅. We claim that there is an open set B1 such that B1 ⊆ V1 ∩ B0 When S is a complete metric space we will require that d(y,x) ≤ 1 when x,y ∈ B1 and when S is locally compact Hausdorff we require that B1 is compact. The argument for this differ in the two cases. Assume first that S is a complete metric space. Take x ∈ V1 ∩ B0. Since V1 ∩ B0 is open there is a 1 > δ > 0 such that {y ∈ S : d(y,x) < δ} ⊆ V1 ∩ B0. Then B1 = y ∈ S : d(y,x) < δ 2 has the desired property. Klaus Thomsen 1. The Baire category theorem (1)
On locally compact Hausdorff spaces Before we continue the proof of Baire’s category theorem we develop a little of the general theory of locally compact Hausdorff spaces. 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 49 and 50: On locally compact Hausdorff spaces
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1. The Baire category theorem - Aarhus Universitet

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