1. The Baire category theorem - Aarhus Universitet

More documents

Recommendations

Info

Proof of Baire’s category theorem - continued 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 Klaus Thomsen 1. The Baire category theorem (3)
Proof of Baire’s category theorem - continued 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. Klaus Thomsen 1. The Baire category theorem (3)
Page 1 and 2:
1. The Baire category theorem Klaus
Page 3 and 4:
We read in W. Rudin: Functional Ana
Page 5 and 6:
Definition of first and second cate
Page 7 and 8:
Page 9 and 10:
Page 11 and 12:
A subset A of S is nowhere dense wh
Page 13 and 14:
A subset A of S is nowhere dense wh
Page 15 and 16:
Page 17 and 18:
Page 19 and 20: Definition of first and second cate
Page 21 and 22: Basic properties the notions of cat
Page 33 and 34: The Baire’s category theorem The
Page 35 and 36: Proof of the corollary Reminder ?:
Page 41 and 42: Proof of Baire’s category theorem
Page 47 and 48: On locally compact Hausdorff spaces
Page 69: On locally compact Hausdorff spaces
Page 89: Proof of Baire’s category theorem
show all

1. The Baire category theorem - Aarhus Universitet

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?