Closure, interior and boundary

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


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]

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


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

[math]\bar{A}=A\cup A'[/math]


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

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


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:



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

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