Closed set

A set that contains every one of its boundary points.

A set $S$ is closed if it contains all of its boundary points -- every point that can be approached arbitrarily closely from both inside and outside $S$ actually belongs to $S$ . Equivalently, a set is closed exactly when its complement (everything not in $S$ ) is open.

Examples

The closed disk $|z|\le 1$ is closed: its entire boundary circle $|z|=1$ is included.
A set can be neither open nor closed, such as $1\le|z|<2$ , which contains part of its boundary but not all of it.