1. The Baire category theorem - Aarhus Universitet

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Proof of Baire’s category theorem - continued 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. This is now straightforward in case of a locally compact Hausdorff space; thanks to the last lemma. Klaus Thomsen 1. The Baire category theorem (3)
Proof of Baire’s category theorem - continued Next we find an open set B2 such that B2 ⊆ V2 ∩ B1 such that d(y,x) ≤ 1 2 when S is complete metric space and such that B2 is compact when S is a locally compact Hausdorff space. Klaus Thomsen 1. The Baire category theorem
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1. The Baire category theorem Klaus
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We read in W. Rudin: Functional Ana
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A subset A of S is nowhere dense wh
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A subset A of S is nowhere dense wh
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1. The Baire category theorem - Aarhus Universitet

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