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Mathematics

Compactness in Topological Spaces and Sequential Compactness

Quick fact

In any metric space, compactness is equivalent to sequential compactness: every sequence has a convergent subsequence. This equivalence is what powers the Bolzano-Weierstrass theorem and the extreme value theorem in calculus.

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You know that a finite set is 'small', but what about an infinite set like the interval [0,1]? It has infinitely many points, yet it behaves 'small' in a way that matters for calculus. What makes it special?

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Compactness in Topological Spaces and Sequential Compactness

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