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