[<< wikibooks] Real Analysis/Limit Points (Accumulation Points)
== Definition ==
Let 
  
    
      
        (
        X
        ,
        d
        )
      
    
    {\displaystyle (X,d)}
   be a metric space, and let 
  
    
      
        A
        ⊂
        X
      
    
    {\displaystyle A\subset X}
  . We call 
  
    
      
        x
        ∈
        X
      
    
    {\displaystyle x\in X}
   a limit point of 
  
    
      
        A
      
    
    {\displaystyle A}
   if for any 
  
    
      
        ϵ
        >
        0
      
    
    {\displaystyle \epsilon >0}
   there exists some 
  
    
      
        y
        ≠
        x
      
    
    {\displaystyle y\neq x}
   such that 
  
    
      
        y
        ∈
        B
        (
        x
        ,
        ϵ
        )
        ∩
        A
      
    
    {\displaystyle y\in B(x,\epsilon )\cap A}
  .
We denote the set 
  
    
      
        l
        i
        m
        (
        A
        )
      
    
    {\displaystyle lim(A)}
   the set of all 
  
    
      
        x
        ∈
        X
      
    
    {\displaystyle x\in X}
   such that 
  
    
      
        x
      
    
    {\displaystyle x}
   is a limit point of 
  
    
      
        A
      
    
    {\displaystyle A}
  .