Chapter 5: Metric Spaces

The limits we've studied so far have been of sequences and function of real numbers. What about sequences in Rⁿ, or functions of several variables? We could treat all of these situations separately, but if we did, we'd find ourselves proving and reproving the same results over and over. So what we do instead, in this chapter, is to develop an abstarct framework that lets us deal with all of the kinds of limits that we'll use. This is called a metric space. In fact it turns out that this context is general enough to lead us to teh useful example of uniform convergence of sequences of functions, which we'll study more in Chapter 9.

Chapter Contents

Class 20
- The complex numbers
- Different types of convergence
Class 21
- The definition of a metric space
- Convergence and continuity in metric spaces
Class 22
Class 23
- Open and closed sets
Class 24
- Images and Preimages
Class 25
- Properties of open and closed sets
- Constructing and recognizing open and closed sets
Class 26
- Sets which are both open and closed
Class 27
- Connected and disconnected sets
Class 28
- Continuous functions and connected sets
  - The logarithm
  - The method of bisection

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: May 1996