Can anyone explain a compact set in a general topological space in an intuitive way?

Question

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!

Answer 1

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.

Answer 2

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.}