07-baire

07-baire - Lecture 7 7 Baire theorem(s In this section we...

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Unformatted text preview: Lecture 7 7. Baire theorem(s) In this section we discuss some topological phenomenon that occurs in certain topological spaces. This deals with interiors of closed sets. Exercise 1 . Let X be a topological space, and let A and B be closed sets with the property that int( A ∪ B ) 6 = ∅ . Prove that either Int( A ) 6 = ∅ , or Int( B ) 6 = ∅ . Exercise 2 . Give an example of a topological space X and of two (non-closed) sets A and B such that Int( A ∪ B ) 6 = ∅ , but Int( A ) = Int( B ) = ∅ . Theorem 7.1 (Baire’s Theorem) . Let ( X, T ) be a topological Hausdorff space, which satisfies one (or both) of the following properties: (a) There exists a meatric d on X , which meakes ( X,d ) a complete metric space, and T is the metric topology. (b) X is locally compact. Suppose one has a sequence ( F n ) n ≥ 1 of closed subsets of X , such that X = S ∞ n =1 F n . Then there exists some integer n ≥ 1 , such that Int( F n ) 6 = ∅ ....
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This note was uploaded on 10/11/2010 for the course MATH 11 taught by Professor Smith during the Three '10 term at ADFA.

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07-baire - Lecture 7 7 Baire theorem(s In this section we...

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