TITLE: How to understand compactness?
QUESTION [19 upvotes]: How to understand the compactness in topology space in intuitive way?

REPLY [7 votes]: The way to understand compactness is to see it in action. As you learn more, you'll see more and more situations in which compactness is useful, even fundamental. With the accumulation of evidence, like geological layers, you will construct understanding. Then one day you'll come across a new way of using compactness, a new angle, and then you will see that in fact you only had understood part of it... and this should keep going.
I am a firm believer that asking for understanding and, much worse, for intuitive understanding of things when one more or less has just encountered them is not the correct way to reach understanding.