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 $\mathrm{int}(A)$. 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. $\mathrm{int}(A)\subseteq A$

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

Title	interior
Canonical name	Interior
Date of creation	2013-03-22 12:48:20
Last modified on	2013-03-22 12:48:20
Owner	yark (2760)
Last modified by	yark (2760)
Numerical id	19
Author	yark (2760)
Entry type	Definition
Classification	msc 54-00
Related topic	Complement
Related topic	Closure
Related topic	BoundaryInTopology
Defines	exterior