Compactness

Compactness - A note on compactness P. B. Kronheimer, for...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

This note was uploaded on 08/30/2011 for the course MATH 3 taught by Professor Other during the Spring '11 term at Adelphi.

Page1 / 2

Compactness - A note on compactness P. B. Kronheimer, for...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online