1. The Baire category theorem - Aarhus Universitet

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Basic properties the notions of category (a) If A ⊆ B and B is of the first category, then so is A. Indeed if B = ∞ i=1 Bi, then A = ∞ i=1 Bi ∩ A and Bi ∩ A ⊆ Bi. (b) Any countable union of sets of the first category is of the first category. Indeed, if B = ∞ i=1 Bi and Bi = ∞ j=1 B(i,j), for all i, then B(i,j), i,j ∈ N is a countable collection of sets whose union is B. (c) Any closed set with empty interior is of the first category. Klaus Thomsen 1. The Baire category theorem
Basic properties the notions of category (a) If A ⊆ B and B is of the first category, then so is A. Indeed if B = ∞ i=1 Bi, then A = ∞ i=1 Bi ∩ A and Bi ∩ A ⊆ Bi. (b) Any countable union of sets of the first category is of the first category. Indeed, if B = ∞ i=1 Bi and Bi = ∞ j=1 B(i,j), for all i, then B(i,j), i,j ∈ N is a countable collection of sets whose union is B. (c) Any closed set with empty interior is of the first category. Indeed, if E is closed, E = E ∪ ∅ ∪ ∅ ∪ ∅ ∪ ... is a union of a countable collection of nowhere dense sets. Klaus Thomsen 1. The Baire category theorem
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 11 and 12: A subset A of S is nowhere dense wh
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Page 21 and 22: Basic properties the notions of cat
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Page 41 and 42: Proof of Baire’s category theorem
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Proof of Baire’s category theorem
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1. The Baire category theorem - Aarhus Universitet

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