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Chapter 4 Compact Sets

You may remember several statements from first and second year analysis that require a closed and bounded interval, for example the Weierstrass extreme value theorem or the Bolzano–Weierstrass theorem. If we want to generalise such statements to metric spaces, then closed and bounded will typically not be sufficient. Instead, we need a concept called compactness. But this comes in two different varieties, and we need another concept to make a connection between them.