I think the definition and motivation of (sequential or open cover) compactness is somewhat unintuitive for beginners. I can find several good discussions on compactness here in math.stackexchange.com. I read them and they were very helpful. However, the compactness is still not so tangible to me. I think it would be helpful if there is a collection of "use cases" of compactness showing its power or usefulness. They could be proofs of some useful theorems involving compact sets or solutions of problems in applied maths or science/engineering. Thank you in advance.
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1See also http://math.stackexchange.com/a/985171/589 and http://math.stackexchange.com/questions/485822/why-is-compactness-so-important. – lhf Oct 28 '16 at 15:24
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You may be interested in the following papers (the links are to JSTOR, hopefully you have access): Hewitt (1960) and Raman-Sundström (2015). I have briefly taken a look at the former, it having been referenced in Sutherland's book, which might also be of interest to you (the introduction to compactness in there seems to be influenced somewhat by Hewitt's paper). I found the second just now while I was looking for the former; it looks at least interesting. I hope these help. – Will R Oct 28 '16 at 19:08