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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
    • Continuous functions are bounded on sequentially compact sets
    • Invertible continuous functions on sequentially compact sets
    • Continuous functions attain their bounds on sequentially compact sets
    • Continued in the next class...
• Class 32
  • Properties of sequentially compact sets (continued)
  • Applications to differentiable functions
  • Continued in the next class...
• 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