Can anyone explain a compact set intuitive way?

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.}