Interior

The interior of a subset $A\subseteq X$ in a topological space $(X,\mathcal{T})$ is

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

A point $x$ lies in $\operatorname{int}(A)$ exactly when $A$ is a neighborhood of $x$. Interior is dual to closure via complements: $\operatorname{int}(A)=X\setminus \overline{X\setminus A}$.

Examples:

• In $\mathbb{R}$ with the usual topology, $\operatorname{int}([0,1])=(0,1)$.
• In $\mathbb{R}$ with the usual topology, $\operatorname{int}(\mathbb{Q})=\varnothing$.
• If $U$ is open, then $\operatorname{int}(U)=U$.