1. The Baire category theorem - Aarhus Universitet

Proof of Baire’s category theorem
Let V1,V2,V3,... be open and dense subsets of S. We must show
that ∞ i=1 Vi is dense in S when S is either a complete metric
space or a locally compact Hausdorff space.
Consider an open non-empty subset B0 of S. We want to show
that ∞ i=1 Vi ∩ B0 = ∅. Since V1 is dense in S, B0 ∩ V1 = ∅.
We claim that there is an open set B1 such that
B1 ⊆ V1 ∩ B0
When S is a complete metric space we will require that d(y,x) ≤ 1
when x,y ∈ B1 and when S is locally compact Hausdorff we require
that B1 is compact.
The argument for this differ in the two cases. Assume first that S
is a complete metric space. Take x ∈ V1 ∩ B0.
Since V1 ∩ B0 is open there is a 1 > δ > 0 such that
{y ∈ S : d(y,x) < δ} ⊆ V1 ∩ B0. Then
B1 = y ∈ S : d(y,x) < δ
2 has the desired property.
Klaus Thomsen 1. The Baire category theorem
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