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
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Page 21 and 22: Basic properties the notions of cat
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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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