interior

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 interiorof $A$ to the be the largest open set contained in $A$.

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

The exterior of $A$ is defined asthe 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. $\mathrm{int}(A)\subseteq A$

2. $\mathrm{int}(A)$ is open

3. $\mathrm{int}(\mathrm{int}(A))=\mathrm{int}(A)$

4. $\mathrm{int}(X)=X$

5. $\mathrm{int}(\mathrm{\varnothing })=\mathrm{\varnothing }$

6. $A$ is open if and only if $A=\mathrm{int}(A)$

7. $\overline{A^{\mathrm{\complement }}}={(\mathrm{int}(A))}^{\mathrm{\complement }}$

8. ${\overline{A}}^{\mathrm{\complement }}=\mathrm{int}(A^{\mathrm{\complement }})$

9. $A\subseteq B$ implies that $\mathrm{int}(A)\subseteq \mathrm{int}(B)$

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

11. $X=\mathrm{int}(A)\cup \partial A\cup \mathrm{int}(A^{\mathrm{\complement }})$