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.
$X=\mathrm{int}(A)\cup \partial A\cup \mathrm{int}({A}^{\mathrm{\complement}})$
References
 1 S. Willard, General Topology, AddisonWesley Publishing Company, 1970.
Title  interior 

Canonical name  Interior 
Date of creation  20130322 12:48:20 
Last modified on  20130322 12:48:20 
Owner  yark (2760) 
Last modified by  yark (2760) 
Numerical id  19 
Author  yark (2760) 
Entry type  Definition 
Classification  msc 5400 
Related topic  Complement 
Related topic  Closure^{} 
Related topic  BoundaryInTopology 
Defines  exterior 