# Closure, interior and boundary

These three things have their own page because of how often they come together.

## Closure

The closure of a set [ilmath]A[/ilmath] denoted [ilmath]\bar{A} [/ilmath] ("bar" in LaTeX) or [ilmath]\overline{A} [/ilmath] ("overline" in LaTeX) is the set:[1]

$\overline{A}=\bigcap\{B\subset X|A\subset B\text{ and }B\text{ is closed in }X\}$

### Alternatives

Alternatively if $A'$ denotes the set of all limit points of $A$ then the closure can be defined as:[2]

$\bar{A}=A\cup A'$

## Interior

The interior of [ilmath]A[/ilmath] denoted by [ilmath]\text{Int }A[/ilmath] or [ilmath]\text{Int}(A)[/ilmath] is the set:

$\text{Int}(A)=\bigcup\{C\subset X|C\subset A\text{ and }C\text{ is open in }X\}$

## Exterior

A less common but still very useful notion is that of exterior, denoted [ilmath]\text{Ext }A[/ilmath] or [ilmath]\text{Ext}(A)[/ilmath] given by:

$\text{Ext}(A)=X-\overline{A}$

## Boundary

The boundary of [ilmath]A[/ilmath], denoted by [ilmath]\partial A[/ilmath] is given by:

$\partial A=X-(\text{Int}(A)\cup\text{Ext}(A))$
1. Introduction to Topological Manifolds - John Lee
2. Walter Rudin - Principals of Mathematical Analysis