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