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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
  Convergence and continuity in Euclidean spaces
• 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