Adherent point

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, an adherent point (also closure point or point of closure or contact point) of a <a href="/facts/Subset/SJGERmbA">subset</a> 
 
 
 
 A
 
 
 {\displaystyle A}
 
 of a <a href="/facts/Topological_space/v2ut7CbK">topological space</a> 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 is a point 
 
 
 
 x
 
 
 {\displaystyle x}
 
 in 
 
 
 
 X
 
 
 {\displaystyle X}
 
 such that every <a href="/facts/Neighbourhood_(mathematics)/XHKgrpkp">neighbourhood</a> of 
 
 
 
 x
 
 
 {\displaystyle x}
 
 (or equivalently, every <a href="/facts/Open_neighborhood/XHKgrpkp">open neighborhood</a> of 
 
 
 
 x
 
 
 {\displaystyle x}
 
) contains at least one point of 
 
 
 
 A
 .
 
 
 {\displaystyle A.}
 
 A point 
 
 
 
 x
 ∈
 X
 
 
 {\displaystyle x\in X}
 
 is an adherent point for 
 
 
 
 A
 
 
 {\displaystyle A}
 
 if and only if 
 
 
 
 x
 
 
 {\displaystyle x}
 
 is in the <a href="/facts/Closure_(topology)/9x2wlgUW">closure</a> of 
 
 
 
 A
 ,
 
 
 {\displaystyle A,}
 
 thus

x
        ∈
        
          Cl
          
            X
          
        
        ⁡
        A
      
    
    {\displaystyle x\in \operatorname {Cl} _{X}A}
  
 if and only if for all open subsets 
  
    
      
        U
        ⊆
        X
        ,
      
    
    {\displaystyle U\subseteq X,}
  
 if 
  
    
      
        x
        ∈
        U
        
           then 
        
        U
        ∩
        A
        ≠
        ∅
        .
      
    
    {\displaystyle x\in U{\text{ then }}U\cap A\neq \varnothing .}

This definition differs from that of a <a href="/facts/Limit_point_of_a_set/95MykoGU">limit point of a set</a>, in that for a limit point it is required that every neighborhood of 
 
 
 
 x
 
 
 {\displaystyle x}
 
 contains at least one point of 
 
 
 
 A
 
 
 {\displaystyle A}
 
 different from 
 
 
 
 x
 .
 
 
 {\displaystyle x.}
 
 Thus every limit point is an adherent point, but the converse is not true. An adherent point of 
 
 
 
 A
 
 
 {\displaystyle A}
 
 is either a limit point of 
 
 
 
 A
 
 
 {\displaystyle A}
 
 or an element of 
 
 
 
 A
 
 
 {\displaystyle A}
 
 (or both). An adherent point which is not a limit point is an <a href="/facts/Isolated_point/d3EeWxuD">isolated point</a>.
Intuitively, having an open set 
 
 
 
 A
 
 
 {\displaystyle A}
 
 defined as the area within (but not including) some boundary, the adherent points of 
 
 
 
 A
 
 
 {\displaystyle A}
 
 are those of 
 
 
 
 A
 
 
 {\displaystyle A}
 
 including the boundary.

Adherent point open-in-new

Adherent point