Let $A$ be a subset of a topological space $X$.

The union of all open sets contained in $A$ is defined to be the interior of $A$. Equivalently, one could define the interior of $A$ to the be the largest open set contained in $A$.

In this entry we denote the interior of $A$ by $\operatorname{int}(A)$. Another common notation is $A^{\circ}$.

The exterior of $A$ is defined as the union of all open sets whose intersection with $A$ is empty. That is, the exterior of $A$ is the interior of the complement of $A$.

The interior of a set enjoys many special properties, some of which are listed below:

1. 1.

   $\operatorname{int}(A)\subseteq A$

2. 2.

   $\operatorname{int}(A)$ is open

3. 3.

   $\operatorname{int}(\operatorname{int}(A))=\operatorname{int}(A)$

4. 4.

   $\operatorname{int}(X)=X$

5. 5.

   $\operatorname{int}(\varnothing)=\varnothing$

6. 6.

   $A$ is open if and only if $A=\operatorname{int}(A)$

7. 7.

   $\overline{A^{\complement}}=(\operatorname{int}(A))^{\complement}$

8. 8.

   $\overline{A}^{\complement}=\operatorname{int}(A^{\complement})$

9. 9.

   $A\subseteq B$ implies that $\operatorname{int}(A)\subseteq\operatorname{int}(B)$

10. 10.

    $\operatorname{int}(A)=A\setminus\partial A$, where $\partial A$ is the boundary of $A$

11. 11.

    $X=\operatorname{int}(A)\cup\partial A\cup\operatorname{int}(A^{\complement})$