Condition under which a set is compact

Question

I'm studying at university real analysis and in class the teacher said that a set is compact if and only if is closed and bounded.

But I don't really understand the concept, more widely: what really means "a set is compact"? And, especially for real analysis, what are the conditions under which a set is compact?

Answer 1

By the general definition,A set is compact if any open cover of the set has a finite subcover. In analysis, you usually deal with euclidean space.

So by heine borel thm,A set is compact in euclidean space iff the set is closed and bounded... Derivation of the heine borel thm can be easily found in any analysis text book