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According to you, what are the deep reasons, at a very fundamental level, that makes the notion of compactness so useful and ubiquitous throughout modern mathematics: theory of compact groups, compact Riemann surfaces, classification of compact topological surfaces, Stone-Weierstrass and Ascoli in analysis... and I'm sure I'm forgetting a dozen of applications of compactness in diverse areas of mathematics..

Somebody told me recently that for him, the basic reason for compactness to be so powerful is that in a sense (which he did not make explicit) a compact space "contains all its infinities". It's a bit like what happens with projective spaces; those spaces are built to "contain all their infinities", which make them more powerful than affine spaces. Would you agree with that ?

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closed as off-topic by Andrés E. Caicedo, Francois Ziegler, Alexandre Eremenko, YCor, Joseph Van Name Feb 10 at 15:51

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    $\begingroup$ math.stackexchange.com/questions/485822/… $\endgroup$ – Asaf Karagila Feb 10 at 8:58
  • $\begingroup$ This is an interesting question, and answers could also touch on logic, on category theory, on sheaf theory (eg Deligne's completeness theorem ncatlab.org/nlab/show/Deligne+completeness+theorem) and so on. But not sure that's what you're looking for, since the analogy to projective spaces seems flawed to me, aside from the fact projective spaces are nice compactifications of affine spaces. $\endgroup$ – David Roberts Feb 10 at 9:10
  • $\begingroup$ The analogy with projective spaces is not mine as I said, and was not clear to me either.. I just mentionned it to ask you guys what do you think of it. The links with logic, category theory or sheafs seems appealing to me indeed, I'd like to know more about it. I will begin to have a look to Deligne's result you mentionned. $\endgroup$ – huurd Feb 10 at 9:31
  • $\begingroup$ In general topology, the quasicompact spaces are precisely the topological spaces where every net (or filter) has an accumulation point. It is quite convenient when your nets always accumulate. $\endgroup$ – Joseph Van Name Feb 10 at 13:34
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    $\begingroup$ Voting to close because already well done at mse. $\endgroup$ – Francois Ziegler Feb 10 at 15:19