Definition – Closed subset of the real numbers.

Definition

Let $A$ be a set in $\mathbb{R}$ . $A$ is said to be closed if it contains its boundry points. Ie, $A$ is closed if $bd~A \subseteq{A}$ .

Notes

It also follows that, for a set $A\subseteq\mathbb{R}$ (do note the possibility of equality), if $A=int~A$ , then it follows that $\mathbb{R}\setminus{A} = \mathbb{R}\setminus{int~A}$ , so that the complement of an open set is closed, and vice versa.