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Unformatted text preview: A note on compactness P. B. Kronheimer, for Math 114 September 1, 2010 These notes are supposed to fill a potential gap, for those who have not seen “compactness” in the context of Math 131 or similar. 1. The BolzanoWeierstrass theorem All the courses that serve as prerequisites for Math 114 deal with the BolzanoWeierstrass theorem, in some form. Here it is, in one common formulation: Theorem 1.1. Every bounded sequence in R d has a convergent subsequence. I won’t write down a proof. Most expositions start with the case d = 1. 2. The HeineBorel theorem Not every course that introduces the BolzanoWeierstrass theorem goes on to cover the “HeineBorel theorem”. (Math 23 has sometimes done this, but not recently, I think.) Here is a statement of the HeineBorel theorem. Theorem 2.1. Let K be a closed and bounded subset of R d , and suppose we have an infinite sequence of open subsets of R d , 2 1 , . . . , 2 n , . . . , whose union covers K: K ⊂ ∞ [ n = 1 2 n . 2 3. Discussion...
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This note was uploaded on 08/30/2011 for the course MATH 3 taught by Professor Other during the Spring '11 term at Adelphi.
 Spring '11
 other
 Math

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