Definition – Open subset of the real numbers.

Definition

Let $A$ be a set in $\mathbb{R}$ . $A$ is said to be open if it does not contain its boundry points. Ie, $A$ is closed if $bd~A \nsubseteq{A}$ .

Notes

It follows from the definition that $A = int~A \Leftrightarrow A$ is closed. To see this, imagine that $A = int~A$ , but that $A$ does include its boundry points. Then, these boundry points would not be a part of the set $int~A$ (by definition). By the definition of the equivalence of two sets, it would be a contradiction for $A = int~A$ , since $A$ would include it’s boundry points while $int~A$ clearly wouldn’t contain $bd~A$ .

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.