Interior - Knowlpedia

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

\operatorname{int}(E):=\bigcup\{\,G\subset E \mid G \text{ is open}\,\}.

Equivalent characterizations

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

Remarks

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 $\operatorname{int}(E)=\emptyset$ .