View source for Closure, interior and boundary

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

==Closure==
The closure of a set {{M|A}} denoted {{M|\bar{A} }} ("bar" in LaTeX) or {{M|\overline{A} }} ("overline" in LaTeX)  is the set:<ref>Introduction to Topological Manifolds - John Lee</ref>

<math>\overline{A}=\bigcap\{B\subset X|A\subset B\text{ and }B\text{ is closed in }X\}</math>

===Alternatives===
Alternatively if <math>A'</math> denotes the set of all [[Limit point|limit points]] of <math>A</math> then the closure can be defined as:<ref>Walter Rudin - Principals of Mathematical Analysis</ref>

<math>\bar{A}=A\cup A'</math>
==Interior==
The interior of {{M|A}} denoted by {{M|\text{Int }A}} or {{M|\text{Int}(A)}} is the set:

<math>\text{Int}(A)=\bigcup\{C\subset X|C\subset A\text{ and }C\text{ is open in }X\}</math>

==Exterior==
A less common but still very useful notion is that of exterior, denoted {{M|\text{Ext }A}} or {{M|\text{Ext}(A)}} given by:

<math>\text{Ext}(A)=X-\overline{A}</math>

==Boundary==
The boundary of {{M|A}}, denoted by {{M|\partial A}} is given by:

<math>\partial A=X-(\text{Int}(A)\cup\text{Ext}(A))</math>

{{Definition|Topology}}