Interior

Let $(X,d)$ be a metric space and let $E\subset X$.

The interior of $E$, denoted $\operatorname{int}(E)$ or $E^\circ$, is defined by

Equivalently, $\operatorname{int}(E)$ is the largest open set contained in $E$.

A pointwise characterization is given by balls inside the set .

Examples:

• In $\mathbb{R}$, $\operatorname{int}([0,1])=(0,1)$.
• If $E$ is open, then $\operatorname{int}(E)=E$.
• If $E$ has empty interior (e.g., the rationals in $\mathbb{R}$), then $\operatorname{int}(E)=\emptyset$.