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

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

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

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

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

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

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

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

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

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

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

# References

- 1
S. Willard,
*General Topology*, Addison-Wesley Publishing Company, 1970.

## Mathematics Subject Classification

54-00*no label found*

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