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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.

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