Interior

Let $(X,d)$ be a metric space and let $A\subseteq X$. The interior of $A$, denoted $\operatorname{int}(A)$ (or $A^\circ$), is the set

(see open ball ).

Equivalently, $\operatorname{int}(A)$ is the union of all open sets contained in $A$. The interior captures the points of $A$ that are not “on the edge” (compare with boundary ).

Examples:

• In $\mathbb{R}$, $\operatorname{int}([0,1])=(0,1)$.
• In $\mathbb{R}^2$, the interior of the closed unit disk $\overline{B}(0,1)$ is the open unit disk $B(0,1)$.
• If $A=\mathbb{Q}\subseteq\mathbb{R}$, then $\operatorname{int}(A)=\varnothing$.