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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Proof of <strong>Baire</strong>’s <strong>category</strong> <strong>theorem</strong> - continued<br />
Next we find an open set B2 such that<br />
B2 ⊆ V2 ∩ B1<br />
such that d(y,x) ≤ 1<br />
2 when S is complete metric space and such<br />
that B2 is compact when S is a locally compact Hausdorff space.<br />
This is done as above: Since V2 is dense, V2 ∩ B1 is not empty.<br />
Repeat then the previous arguments.<br />
Thus we construct recursively a sequence B0,B1,B2,... of open<br />
sets such that<br />
Bn ⊆ Vn ∩ Bn−1<br />
for all n ≥ 1, d(x,y) ≤ 1<br />
n when x,y ∈ Bn in the case where S is a<br />
complete metric space, and such that Bn is compact in the case<br />
where S is a locally compact Hausdorff space.<br />
Klaus Thomsen <strong>1.</strong> <strong>The</strong> <strong>Baire</strong> <strong>category</strong> <strong>theorem</strong>