Editing Boundary (topology) (section)

== Properties ==

The closure of a set <math>S</math> equals the union of the set with its boundary: 
<math display="block">\overline{S} = S \cup \partial_X S</math> 
where <math>\overline{S} = \operatorname{cl}_X S</math> denotes the [[Closure (topology)|closure]] of <math>S</math> in <math>X.</math> 
A set is closed if and only if it contains its boundary, and open if and only if it is disjoint from its boundary. The boundary of a set is [[Closed set|closed]];<ref>{{cite book|last=Mendelson|first=Bert|date=1990|orig-year=1975|title=Introduction to Topology|edition=Third|publisher=Dover|isbn=0-486-66352-3|page=86|quote=Corollary 4.15 For each subset <math>A,</math> <math>\operatorname{Bdry} (A)</math> is closed.}}</ref> this follows from the formula <math>\partial_X S ~:=~ \overline{S} \cap \overline{(X \setminus S)},</math> which expresses <math>\partial_X S</math> as the intersection of two closed subsets of <math>X.</math> 

("Trichotomy"){{Anchor|Trichotomy}}<!-- Linked to from [[Nowhere dense set]] --> Given any subset <math>S \subseteq X,</math> each point of <math>X</math> lies in exactly one of the three sets <math>\operatorname{int}_X S, \partial_X S,</math> and <math>\operatorname{int}_X (X \setminus S).</math>  Said differently, <math display="block">X ~=~ \left(\operatorname{int}_X S\right) \;\cup\; \left(\partial_X S\right) \;\cup\; \left(\operatorname{int}_X (X \setminus S)\right)</math> and these three sets are [[pairwise disjoint]]. Consequently, if these set are not empty<ref group=note>The condition that these sets be non-empty is needed because sets in a [[Partition of a set|partition]] are by definition required to be non-empty.</ref> then they form a [[Partition of a set|partition]] of <math>X.</math> 

A point <math>p \in X</math> is a boundary point of a set if and only if every neighborhood of <math>p</math> contains at least one point in the set and at least one point not in the set. 
The boundary of the interior of a set as well as the boundary of the closure of a set are both contained in the boundary of the set. 

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[[File:Accumulation And Boundary Points Of S.PNG]]<br/>
''Conceptual [[Venn diagram]] showing the relationships among different points of a subset <math>S</math> of <math>\R^n.</math> <math>A</math> = set of [[accumulation point]]s of <math>S</math> (also called limit points), <math>B = </math> set of '''boundary points''' of <math>S,</math> area shaded green = set of [[interior point]]s of <math>S,</math> area shaded yellow = set of [[isolated point]]s of <math>S,</math> areas shaded black = empty sets.  Every point of <math>S</math> is either an interior point or a boundary point.  Also, every point of <math>S</math> is either an accumulation point or an isolated point.  Likewise, every boundary point of <math>S</math> is either an accumulation point or an isolated point.  Isolated points are always boundary points.''
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