Adherent point

<h2 id="examples-and-sufficient-conditions">Examples and sufficient conditions</h2>
<p>If 
  
    
      
        S
      
    
    {\displaystyle S}
  
 is a <a href="/facts/Empty_set/ffEk3eA6">non-empty</a> subset of 
  
    
      
        
          R
        
      
    
    {\displaystyle \mathbb {R} }
  
 which is bounded above, then the <a href="/facts/Supremum/oXCKVopz">supremum</a> 
  
    
      
        sup
        S
      
    
    {\displaystyle \sup S}
  
 is adherent to 
  
    
      
        S
        .
      
    
    {\displaystyle S.}
  
 In the <a href="/facts/Interval_(mathematics)/e9se3vTe">interval</a> 
  
    
      
        (
        a
        ,
        b
        ]
        ,
      
    
    {\displaystyle (a,b],}
  
 
  
    
      
        a
      
    
    {\displaystyle a}
  
 is an adherent point that is not in the interval, with usual <a href="/facts/Topological_space/v2ut7CbK">topology</a> of 
  
    
      
        
          R
        
        .
      
    
    {\displaystyle \mathbb {R} .}

</p><p>A subset 
  
    
      
        S
      
    
    {\displaystyle S}
  
 of a <a href="/facts/Metric_space/gnBBRXtc">metric space</a> 
  
    
      
        M
      
    
    {\displaystyle M}
  
 contains all of its adherent points if and only if 
  
    
      
        S
      
    
    {\displaystyle S}
  
 is (<a href="/facts/Sequential_space/xAjFpdfR">sequentially</a>) <a href="/facts/Closed_set/upYnRTUj">closed</a> in 
  
    
      
        M
        .
      
    
    {\displaystyle M.}

</p>
<h3>Adherent points and subspaces</h3>
<p>Suppose 
  
    
      
        x
        ∈
        X
      
    
    {\displaystyle x\in X}
  
 and 
  
    
      
        S
        ⊆
        X
        ⊆
        Y
        ,
      
    
    {\displaystyle S\subseteq X\subseteq Y,}
  
 where 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is a <a href="/facts/Topological_subspace/NjU6z4Ca">topological subspace</a> of 
  
    
      
        Y
      
    
    {\displaystyle Y}
  
 (that is, 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is endowed with the <a href="/facts/Subspace_topology/NjU6z4Ca">subspace topology</a> induced on it by 
  
    
      
        Y
      
    
    {\displaystyle Y}
  
). Then 
  
    
      
        x
      
    
    {\displaystyle x}
  
 is an adherent point of 
  
    
      
        S
      
    
    {\displaystyle S}
  
 in 
  
    
      
        X
      
    
    {\displaystyle X}
  
 if and only if 
  
    
      
        x
      
    
    {\displaystyle x}
  
 is an adherent point of 
  
    
      
        S
      
    
    {\displaystyle S}
  
 in 
  
    
      
        Y
        .
      
    
    {\displaystyle Y.}

</p>

<table><tbody><tr><th>Proof</th></tr><tr><td><p>By assumption,                     S        ⊆        X        ⊆        Y              {\displaystyle S\subseteq X\subseteq Y}   and                     x        ∈        X        .              {\displaystyle x\in X.}   Assuming that                     x        ∈                  Cl                      X                          ⁡        S        ,              {\displaystyle x\in \operatorname {Cl} _{X}S,}   let                     V              {\displaystyle V}   be a neighborhood of                     x              {\displaystyle x}   in                     Y              {\displaystyle Y}   so that                     x        ∈                  Cl                      Y                          ⁡        S              {\displaystyle x\in \operatorname {Cl} _{Y}S}   will follow once it is shown that                     V        ∩        S        ≠        ∅        .              {\displaystyle V\cap S\neq \varnothing .}   The set                     U        :=        V        ∩        X              {\displaystyle U:=V\cap X}   is a neighborhood of                     x              {\displaystyle x}   in                     X              {\displaystyle X}   (by definition of the <a href="/facts/Subspace_topology/NjU6z4Ca">subspace topology</a>) so that                     x        ∈                  Cl                      X                          ⁡        S              {\displaystyle x\in \operatorname {Cl} _{X}S}   implies that                     ∅        ≠        U        ∩        S        .              {\displaystyle \varnothing \neq U\cap S.}   Thus                     ∅        ≠        U        ∩        S        =        (        V        ∩        X        )        ∩        S        ⊆        V        ∩        S        ,              {\displaystyle \varnothing \neq U\cap S=(V\cap X)\cap S\subseteq V\cap S,}   as desired. For the converse, assume that                     x        ∈                  Cl                      Y                          ⁡        S              {\displaystyle x\in \operatorname {Cl} _{Y}S}   and let                     U              {\displaystyle U}   be a neighborhood of                     x              {\displaystyle x}   in                     X              {\displaystyle X}   so that                     x        ∈                  Cl                      X                          ⁡        S              {\displaystyle x\in \operatorname {Cl} _{X}S}   will follow once it is shown that                     U        ∩        S        ≠        ∅        .              {\displaystyle U\cap S\neq \varnothing .}   By definition of the <a href="/facts/Subspace_topology/NjU6z4Ca">subspace topology</a>, there exists a neighborhood                     V              {\displaystyle V}   of                     x              {\displaystyle x}   in                     Y              {\displaystyle Y}   such that                     U        =        V        ∩        X        .              {\displaystyle U=V\cap X.}   Now                     x        ∈                  Cl                      Y                          ⁡        S              {\displaystyle x\in \operatorname {Cl} _{Y}S}   implies that                     ∅        ≠        V        ∩        S        .              {\displaystyle \varnothing \neq V\cap S.}   From                     S        ⊆        X              {\displaystyle S\subseteq X}   it follows that                     S        =        X        ∩        S              {\displaystyle S=X\cap S}   and so                     ∅        ≠        V        ∩        S        =        V        ∩        (        X        ∩        S        )        =        (        V        ∩        X        )        ∩        S        =        U        ∩        S        ,              {\displaystyle \varnothing \neq V\cap S=V\cap (X\cap S)=(V\cap X)\cap S=U\cap S,}   as desired.                     ◼              {\displaystyle \blacksquare }  </p></td></tr></tbody></table>
<p>Consequently, 
  
    
      
        x
      
    
    {\displaystyle x}
  
 is an adherent point of 
  
    
      
        S
      
    
    {\displaystyle S}
  
 in 
  
    
      
        X
      
    
    {\displaystyle X}
  
 if and only if this is true of 
  
    
      
        x
      
    
    {\displaystyle x}
  
 in every (or alternatively, in some) topological superspace of 
  
    
      
        X
        .
      
    
    {\displaystyle X.}

</p>
<h3>Adherent points and sequences</h3>
<p>If 
  
    
      
        S
      
    
    {\displaystyle S}
  
 is a subset of a topological space then the <a href="/facts/Limit_of_a_sequence/Ktge2RwL">limit</a> of a convergent sequence in 
  
    
      
        S
      
    
    {\displaystyle S}
  
 does not necessarily belong to 
  
    
      
        S
        ,
      
    
    {\displaystyle S,}
  
 however it is always an adherent point of 
  
    
      
        S
        .
      
    
    {\displaystyle S.}
  
 Let 
  
    
      
        
          
            (
            
              x
              
                n
              
            
            )
          
          
            n
            ∈
            
              N
            
          
        
      
    
    {\displaystyle \left(x_{n}\right)_{n\in \mathbb {N} }}
  
 be such a sequence and let 
  
    
      
        x
      
    
    {\displaystyle x}
  
 be its limit. Then by definition of limit, for all <a href="/facts/Neighbourhood_(mathematics)/XHKgrpkp">neighbourhoods</a> 
  
    
      
        U
      
    
    {\displaystyle U}
  
 of 
  
    
      
        x
      
    
    {\displaystyle x}
  
 there exists 
  
    
      
        n
        ∈
        
          N
        
      
    
    {\displaystyle n\in \mathbb {N} }
  
 such that 
  
    
      
        
          x
          
            n
          
        
        ∈
        U
      
    
    {\displaystyle x_{n}\in U}
  
 for all 
  
    
      
        n
        ≥
        N
        .
      
    
    {\displaystyle n\geq N.}
  
 In particular, 
  
    
      
        
          x
          
            N
          
        
        ∈
        U
      
    
    {\displaystyle x_{N}\in U}
  
 and also 
  
    
      
        
          x
          
            N
          
        
        ∈
        S
        ,
      
    
    {\displaystyle x_{N}\in S,}
  
 so 
  
    
      
        x
      
    
    {\displaystyle x}
  
 is an adherent point of 
  
    
      
        S
        .
      
    
    {\displaystyle S.}
  
 
In contrast to the previous example, the limit of a convergent sequence in 
  
    
      
        S
      
    
    {\displaystyle S}
  
 is not necessarily a limit point of 
  
    
      
        S
      
    
    {\displaystyle S}
  
; for example consider 
  
    
      
        S
        =
        {
        0
        }
      
    
    {\displaystyle S=\{0\}}
  
 as a subset of 
  
    
      
        
          R
        
        .
      
    
    {\displaystyle \mathbb {R} .}
  
 Then the only sequence in 
  
    
      
        S
      
    
    {\displaystyle S}
  
 is the constant sequence 
  
    
      
        0
        ,
        0
        ,
        …
      
    
    {\displaystyle 0,0,\ldots }
  
 whose limit is 
  
    
      
        0
        ,
      
    
    {\displaystyle 0,}
  
 but 
  
    
      
        0
      
    
    {\displaystyle 0}
  
 is not a limit point of 
  
    
      
        S
        ;
      
    
    {\displaystyle S;}
  
 it is only an adherent point of 
  
    
      
        S
        .
      
    
    {\displaystyle S.}

</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Closed_set/upYnRTUj">Closed set</a> – Complement of an open subset</li>
<li><a href="/facts/Closure_(topology)/9x2wlgUW">Closure (topology)</a> – All points and limit points in a subset of a topological space</li>
<li><a href="/facts/Limit_of_a_sequence/Ktge2RwL">Limit of a sequence</a> – Value to which tends an infinite sequence</li>
<li><a href="/facts/Limit_point_of_a_set/95MykoGU">Limit point of a set</a> – Cluster point in a topological spacePages displaying short descriptions of redirect targets</li>
<li><a href="/facts/Subsequential_limit/SHI6KoMy">Subsequential limit</a> – The limit of some subsequence</li></ul>
<h2 id="notes">Notes</h2>

<h2 id="citations">Citations</h2>

<ul><li>Adamson, Iain T., <i><a href="https://books.google.com/books?id=CChCVUMrQzgC&q=%22Adherent+point%22">A General Topology Workbook</a></i>, Birkhäuser Boston; 1st edition (November 29, 1995). <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8176-3844-3.</li>
<li><a href="/facts/Tom_M._Apostol/S6vGtJWN">Apostol, Tom M.</a>, <i>Mathematical Analysis</i>, Addison Wesley Longman; second edition (1974). <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-201-00288-4</li>
<li><a href="/facts/Seymour_Lipschutz/lpf9648a">Lipschutz, Seymour</a>; <i>Schaum's Outline of General Topology</i>, McGraw-Hill; 1st edition (June 1, 1968). <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-07-037988-2.</li>
<li><a href="/facts/Lynn_Steen/JFdvAfpc">L.A. Steen</a>, <a href="/facts/J._Arthur_Seebach%2c_Jr./sU4TbeUA">J.A.Seebach, Jr.</a>, <i>Counterexamples in topology</i>, (1970) Holt, Rinehart and Winston, Inc..</li>
<li><i>This article incorporates material from <a href="https://planetmath.org/adherentpoint">Adherent point</a> on <a href="/facts/PlanetMath/wphPCtLf">PlanetMath</a>, which is licensed under the Creative Commons Attribution/Share-Alike License.</i></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Steen, p. 5; Lipschutz, p. 69; Adamson, p. 15. <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
</ol>

Adherent point open-in-new

Adherent point