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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