Compact set

Closed AND bounded -- the property that guarantees an absolute max and min exist.

In $\mathbb{R}^n$ , a set is compact exactly when it is both closed and bounded (Heine--Borel). Compactness is the hypothesis the multivariable Extreme Value Theorem needs to guarantee a continuous function actually attains an absolute maximum and minimum on the set.

Examples

A closed disk $x^2+y^2\leq4$ is compact; the open disk $x^2+y^2<4$ is bounded but not compact, since it's missing its boundary.
The whole plane $\mathbb{R}^2$ is closed but not bounded, so it isn't compact either.