View source for Adherent point

{{short description|Point that belongs to the closure of some given subset of a topological space}}
In [[Mathematics|mathematics]], an '''adherent point''' (also '''closure point''' or '''point of closure''' or '''contact point''')<ref>Steen, p. 5; Lipschutz, p. 69; Adamson, p. 15.</ref> of a [[Subset|subset]] <math>A</math> of  a [[Topological space|topological space]] <math>X,</math> is a point <math>x</math> in <math>X</math> such that every [[Neighbourhood (mathematics)|neighbourhood]] of <math>x</math> (or equivalently, every open neighborhood of <math>x</math>) contains at least one point of <math>A.</math> A point <math>x \in X</math> is an adherent point for <math>A</math> if and only if <math>x</math> is in the [[Closure (topology)|closure]] of <math>A,</math> thus
:<math>x \in \operatorname{Cl}_X A</math> if and only if for all open subsets <math>U \subseteq X,</math> if <math>x \in U \text{ then } U \cap A \neq \varnothing.</math>

This definition differs from that of a [[Limit point of a set|limit point of a set]], in that for a limit point it is required that every neighborhood of <math>x</math> contains at least one point of <math>A</math> {{em|different from}} <math>x.</math> Thus every limit point is an adherent point, but the converse is not true. An adherent point of <math>A</math> is either a limit point of <math>A</math> or an element of <math>A</math> (or both). An adherent point which is not a limit point is an [[Isolated point|isolated point]].

Intuitively, having an open set <math>A</math> defined as the area within (but not including) some boundary, the adherent points of <math>A</math> are those of <math>A</math> including the boundary.

==Examples and sufficient conditions==

If <math>S</math> is a [[Empty set|non-empty]] subset of <math>\R</math> which is bounded above, then the supremum <math>\sup S</math> is adherent to <math>S.</math> In the [[Interval (mathematics)|interval]] <math>(a, b],</math> <math>a</math> is an adherent point that is not in the interval, with usual [[Topological space|topology]] of <math>\R.</math>

A subset <math>S</math> of a [[Metric space|metric space]] <math>M</math> contains all of its adherent points if and only if <math>S</math> is ([[Sequential space#Definitions|sequentially]]) [[Closed set|closed]] in <math>M.</math>

===Adherent points and subspaces===

Suppose <math>x \in X</math> and <math>S \subseteq X \subseteq Y,</math> where <math>X</math> is a topological subspace of <math>Y</math> (that is, <math>X</math> is endowed with the [[Subspace topology|subspace topology]] induced on it by <math>Y</math>). Then <math>x</math> is an adherent point of <math>S</math> in <math>X</math> if and only if <math>x</math> is an adherent point of <math>S</math> in <math>Y.</math>

{{Collapse top|left=yes|title=Proof}}
By assumption, <math>S \subseteq X \subseteq Y</math> and <math>x \in X.</math> Assuming that <math>x \in \operatorname{Cl}_X S,</math> let <math>V</math> be a neighborhood of <math>x</math> in <math>Y</math> so that <math>x \in \operatorname{Cl}_Y S</math> will follow once it is shown that <math>V \cap S \neq \varnothing.</math> The set <math>U := V \cap X</math> is a neighborhood of <math>x</math> in <math>X</math> (by definition of the [[Subspace topology|subspace topology]]) so that <math>x \in \operatorname{Cl}_X S</math> implies that <math>\varnothing \neq U \cap S.</math> Thus <math>\varnothing \neq U \cap S = (V \cap X) \cap S \subseteq V \cap S,</math> as desired. For the converse, assume that <math>x \in \operatorname{Cl}_Y S</math> and let <math>U</math> be a neighborhood of <math>x</math> in <math>X</math> so that <math>x \in \operatorname{Cl}_X S</math> will follow once it is shown that <math>U \cap S \neq \varnothing.</math> By definition of the [[Subspace topology|subspace topology]], there exists a neighborhood <math>V</math> of <math>x</math> in <math>Y</math> such that <math>U = V \cap X.</math> Now <math>x \in \operatorname{Cl}_Y S</math> implies that <math>\varnothing \neq V \cap S.</math> From <math>S \subseteq X</math> it follows that <math>S = X \cap S</math> and so <math>\varnothing \neq V \cap S = V \cap (X \cap S) = (V \cap X) \cap S = U \cap S,</math> as desired. <math>\blacksquare</math>
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Consequently, <math>x</math> is an adherent point of <math>S</math> in <math>X</math> if and only if this is true of <math>x</math> in every (or alternatively, in some) topological superspace of <math>X.</math>

===Adherent points and sequences===

If <math>S</math> is a subset of a topological space then the [[Limit (mathematics)#In topology|limit]] of a convergent sequence in <math>S</math> does not necessarily belong to <math>S,</math> however it is always an adherent point of <math>S.</math> Let <math>\left(x_n\right)_{n \in \N}</math> be such a sequence and let <math>x</math> be its limit. Then by definition of limit, for all [[Neighbourhood (mathematics)|neighbourhood]]s <math>U</math> of <math>x</math> there exists <math>n \in \N</math> such that <math>x_n \in U</math> for all <math>n \geq N.</math> In particular, <math>x_N \in U</math> and also <math>x_N \in S,</math> so <math>x</math> is an adherent point of <math>S.</math> 
In contrast to the previous example, the limit of a convergent sequence in <math>S</math> is not necessarily a limit point of <math>S</math>; for example consider <math>S = \{ 0 \}</math> as a subset of <math>\R.</math> Then the only sequence in <math>S</math> is the constant sequence <math>0, 0, \ldots</math> whose limit is <math>0,</math> but <math>0</math> is not a limit point of <math>S;</math> it is only an adherent point of <math>S.</math>

==See also==

* {{annotated link|Closed set}}
* {{annotated link|Closure (topology)}}
* {{annotated link|Limit of a sequence}}
* {{annotated link|Limit point of a set}}
* {{annotated link|Subsequential limit}}

==Notes==

{{reflist|group=note}}
{{reflist|group=proof}}

==Citations==

{{reflist}}

==References==

* Adamson, Iain T., ''[https://books.google.com/books?id=CChCVUMrQzgC&q=%22Adherent+point%22 A General Topology Workbook]'', Birkhäuser Boston; 1st edition (November 29, 1995). {{ISBN|978-0-8176-3844-3}}.
* [[Biography:Tom M. Apostol|Apostol, Tom M.]], ''Mathematical Analysis'', Addison Wesley Longman; second edition (1974). {{ISBN|0-201-00288-4}}
* Lipschutz, Seymour; ''Schaum's Outline of General Topology'', McGraw-Hill; 1st edition (June 1, 1968). {{ISBN|0-07-037988-2}}. 
* L.A. Steen, J.A.Seebach, Jr., ''Counterexamples in topology'', (1970) Holt, Rinehart and Winston, Inc.. 

[[Category:General topology]]

{{Sourceattribution|Adherent point}}