1. The Baire category theorem - Aarhus Universitet

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On locally compact Hausdorff spaces Proof. Consider an element k ∈ K. Since X is Hausdorff there are open sets, Wk ⊆ X and Uk ⊆ X, such that k ∈ Wk,p ∈ Uk and Wk ∩ Uk = ∅. Note that the collection {Wk : k ∈ K} is an open cover of K. Since K is compact there is a finite collection k1,k2,... ,kN such that K ⊆ Wk1 ∪ Wk2 ∪ · · · ∪ WkN . Set W = Wk1 ∪ Wk2 ∪ · · · ∪ WkN and U = Uk1 ∩ Uk2 ∩ · · · ∩ UkN . Corollary: In a Hausdorff space, any compact subset is closed. Proof: By Lemma1 the complement of a compact set is open. 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 = ∅. Klaus Thomsen 1. The Baire category theorem
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1. The Baire category theorem Klaus
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1. The Baire category theorem - Aarhus Universitet

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