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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 Bolzano-Weierstrass theorem All the courses that serve as prerequisites for Math 114 deal with the Bolzano-Weierstrass 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 Heine-Borel theorem Not every course that introduces the Bolzano-Weierstrass theorem goes on to cover the “Heine-Borel theorem”. (Math 23 has sometimes done this, but not recently, I think.) Here is a statement of the Heine-Borel 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