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I know that there are many ways to define the compactness of the set in a topological space, e.g., $X$ is compact if and only if every open cover of $X$ has a finite subcover. I also know that it is the extension of the concept of the bounded closed set in an Euclidean space to a general topological space.

However, I do not believe I truly understand what it is intuitively. Can anyone help me grasp the concept intuitively?

** I already got a wonderful answer, but please add more answers if you have different (or even similar) insight of yours. I can't examine what more I could learn on this post. I'm very excited!

Sunghee Yun
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    You know how on closed and bounded sets in the reals, you have the extreme value, intermediate value theorems? That's the intuition you should have for other normal spaces and compact subsets. – Rushabh Mehta Feb 16 '20 at 05:15
  • @DonThousand Interesting! But, I now why extreme value theorem and intermediate value theorem hold. The proof depends on Bolzano-Weierstrass theorem, whose proof depends on monotone convergence theorem, whose proof depends on least-upper-bound property, whose proof depends on the completeness axiom for real numbers. That is, essentially those theorem hold only because of the axiom we have for real numbers. I think I have the intuition there. However, I still don't see how that's related to compact subsets. Can you show me how? – Sunghee Yun Feb 16 '20 at 05:34
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    The theorems ONLY hold in the reals over compact sets. Bolzano Weierstrauss is equivalent to sequential compactness, for example. – Rushabh Mehta Feb 16 '20 at 16:31
  • @DonThousand Interesting! Could you be more specific about how my intuition in real space help me reach understanding of compactness in general topological space? I'd greatly appreciate if you create an answer instead of comments. :) – Sunghee Yun Feb 16 '20 at 18:50

2 Answers2

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This paraphrase of the finite subcover definition of compactness is atributed to Hermann Weyl:

"If a city is compact it can be guarded by a finite number of arbitrarily near-sighted policemen".

It's clear that this characterisation is trivially true for finite sets and actually on first sight you might be tempted to think that it is only true for finite sets since it appears to be quite a strong condition. Hence a key observation concerning compactness is that it is a non-trivial generalisation of finiteness.

In Edwin Hewitt's Essay, "The rôle of compactness in analysis" he says that:

"The thesis of this essay is that a great many propositions of analysis are:

  1. trivial for finite sets.
  2. true and reasonably simple for infinite compact sets.
  3. either false or extremely difficult to prove for noncompact sets."

*Hewitt, Edwin, The rôle of compactness in analysis, Am. Math. Mon. 67, 499-516 (1960). ZBL0101.15302.

Ivan
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Compact sets have many nice properties of finite sets but they can be infinite. For example, in finite sets:

  • All functions have a maximum
  • All functions are bounded
  • All sequences have a constant subsequence

In compact sets:

  • All continuous functions have a maximum
  • All continuous functions are bounded
  • Spaces where all subsequences have a convergent subsequence are called sequentially compact

Just like finite sets but infinite.

jgon
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Vicfred
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    It would be nice if the downvoter would explain – J. W. Tanner Feb 16 '20 at 05:38
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    Maybe my humble answer has not enough fancy words for his taste – Vicfred Feb 16 '20 at 05:40
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    Corrections: "A constant subsequence" is false. Even a convergent subsequence is false in general, but holds in metric spaces, e.g. "Functions" should say "continuous functions" of course. – Henno Brandsma Feb 16 '20 at 06:59
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    @Vicfred Thanks for your answer! Probably due to my ignorance, I could not get much insight, but your answer definitely helped me understand the subject better. Thanks again! – Sunghee Yun Feb 25 '20 at 04:06