Closed set

A set that contains all of its own boundary points.

A set $F\subseteq\mathbb{R}^n$ is closed if it contains every one of its boundary points -- equivalently, if its complement is open. A closed set can still fail to be "small" or bounded: $\mathbb{R}^n$ itself and any single point are both closed.

Examples

The closed disk $\{(x,y): x^2+y^2\leq1\}$ is closed -- it contains its own boundary circle.
A finite set of points is always closed, since every one of its points is (trivially) a boundary point it contains.