compactness
Compactness is topology's way of capturing the idea of a space being 'finite-ish' — bounded and complete, with nothing leaking off to infinity and no points missing from the edge. A closed solid ball is compact; the infinite line is not (it runs off forever); an open interval (0, 1) is not (its endpoints are missing, so a sequence can creep toward 0 with no limit inside). Intuitively, a compact space is one you can never escape from by sliding off an edge or running to infinity — wherever you try to drift, you are caught.
The clean modern definition is about covers. Suppose you blanket the space with a collection of open sets so that every point lies in at least one of them — an 'open cover'. The space is compact if, no matter what open cover you start with, you can always keep just finitely many of those sets and still cover everything: every open cover has a finite subcover. This sounds abstract, but it is exactly the property that lets you turn infinitely many local facts into one global fact. On the line and in ordinary n-dimensional space there is a friendly shortcut, the Heine-Borel theorem: a subset is compact exactly when it is closed and bounded.
Compactness is one of the most useful invariants because it guarantees good behaviour: a continuous real-valued function on a compact space always attains a maximum and a minimum (the extreme value theorem), and the continuous image of a compact space is compact. It is also how topology certifies a surface is 'finite' — a sphere and a torus are compact, an infinite plane is not. The honest subtlety: 'closed and bounded means compact' is true in ordinary finite-dimensional space but fails in general (in infinite-dimensional spaces a closed bounded set can fail to be compact), so the open-cover definition is the real one and Heine-Borel is a special, comfortable case.
Cover the closed interval [0, 1] with all the open intervals of the form (1/n - 0.01, 1) for n = 1, 2, 3, ... together with one small open set around 0. This is an open cover with infinitely many pieces, yet you can pick out just a finite handful that still cover everything from 0 to 1 — because [0, 1] is compact. Try the same trick on the open interval (0, 1] using (1/n, 2): no finite subcollection reaches all the way down to 0, so (0, 1] is not compact.
A closed bounded interval is compact; drop the closed endpoint and the finite-subcover guarantee breaks.
'Closed and bounded means compact' (Heine-Borel) is a feature of finite-dimensional Euclidean space only; in general the open-cover definition is the genuine one, and a closed bounded set can fail to be compact in infinite dimensions.