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 $\int(A)$ 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. $\int(A)\subseteq A$
2. $\int(A)$ is open
3. $\int(\int(A))=\int(A)$
4. $\int(X)=X$
5. $\int(\emptyset)=\emptyset$
6. $A$ is open if and only if $A=\int(A)$
7. $\overline{A^\complement}=(\int(A))^\complement$
8. $\overline{A}^\complement = \int(A^\complement)$
9. $A\subseteq B$ implies that $\int(A)\subseteq \int(B)$
10. $\int(A)=A\setminus \partial A$ where $\partial A$ is the boundary of $A$
11. $X=\int(A)\cup \partial A \cup \int(A^\complement)$

Bibliography

1

S. Willard, General Topology, Addison-Wesley Publishing Company, 1970.