Interior - Knowlpedia

Empty set and whole space: ∅ , X ∈ T \varnothing,X\in\mathcal T ∅ , X ∈ T .
Arbitrary unions: the union of any collection of members of T \mathcal T T belongs to T \mathcal T T .
Finite intersections: the intersection of finitely many members of T \mathcal T T belongs to T \mathcal T T .

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

\operatorname{int}(A)=\bigcup\{U\in\mathcal{T}: U\subseteq A\}.

Equivalent characterizations

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

Remarks

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 $\operatorname{int}(U)=U$ .