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I recently picked one of my favorite math books out of my closet and started flipping through it again (Gamelin & Greene's Intro to Topology).

I'm continuing through the second half now, which is on algebraic topology. But there was one thing that I never quite "got" in point-set topology, and that's the intuition behind compactness.

I understand the definition. A compact set is a set where every cover has a finite subcover. I understand it's equivalent to being closed and bounded in R^n. But what I don't quite get it why its important.

I've heard the saying "compactness is the next best thing to finiteness". It sounds like this hints at an underlying motivation for the property. Maybe it might help to elaborate on this quote.

Sorry for the unruly open-ended question =-)

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# Intuition behind compactness

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