Chapter 6: Compact Sets

Compactness is a tremendously useful property that some (the very nicest) metric spaces have. Continuous functions, in particular, are far better behaved in compact metric spaces than any others. The results we prove in this chapter on the relationship between continuous functions and compact metric spaces lead to some of the most important and deepest results of this course. Powerful results such as Rolle's Theorem, the Mean Value Theorem and the Inverse function Theorem all come as fairly straightforward applications of the results on compactness that we prove in the first part of the chapter. It's no exaggeration to say this is the core chapter of the course.

Chapter Contents

Class 29
- Subsequences
Class 30
- Sequentially compact sets
Class 31
- Properties of sequentially compact sets
Class 32
Class 33
- Uniform Continuity
Class 34
- Compact Sets
Class 35
- Compactness and Sequential Compactness are Equivalent
  - Compactness and Sequential Compactness
  - Countable and Uncountable sets
Class 36
- Compactness and Sequential Compactness are Equivalent
- On with the proof of Theorem 6.24
Class 37
- Completeness

Analysis WebNotes by John Lindsay Orr.
Comments to the author: jorr@math.unl.edu

All contents copyright (C) 1996 John L. Orr
University of Nebraska--Lincoln
All rights reserved

Last modified: June 1996