1. The Baire category theorem - Aarhus Universitet

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On locally compact Hausdorff spaces Lemma Let X be a Hausdorff space and {Kα : α ∈ A} a collection of compact subsets of X. If α∈A Kα = ∅, there is a finite subset F ⊆ A such that α∈F Kα = ∅. Proof. Fix one of the Kα’s; say Kα1 . Since α∈A Kα = ∅, every element of Kα1 is in the complement Kc α of Kα, for some α. This means that {Kc α : α ∈ A} covers Kα1 . Klaus Thomsen 1. The Baire category theorem
On locally compact Hausdorff spaces Lemma Let X be a Hausdorff space and {Kα : α ∈ A} a collection of compact subsets of X. If α∈A Kα = ∅, there is a finite subset F ⊆ A such that α∈F Kα = ∅. Proof. Fix one of the Kα’s; say Kα1 . Since α∈A Kα = ∅, every element of Kα1 is in the complement Kc α of Kα, for some α. This means that {Kc α : α ∈ A} covers Kα1 . By Corollary ?? each Kc α is open, so by compactness of Kα1 there is a finite subset F0 ⊆ A such that ⊆ K c α. Kα1 α∈F0 Set F = {α1} ∪ F0. Then α∈F Kα = ∅. 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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1. The Baire category theorem - Aarhus Universitet

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