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. Klaus Thomsen 1. The Baire category theorem (1)
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. Klaus Thomsen 1. The Baire category theorem (1)
Page 1 and 2: 1. The Baire category theorem Klaus
Page 3 and 4: We read in W. Rudin: Functional Ana
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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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