First-countable space

<p>In <a href="/facts/Topology/2EqW47cU">topology</a>, a branch of <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a first-countable space is a <a href="/facts/Topological_space/v2ut7CbK">topological space</a> satisfying the "first <a href="/facts/Axiom_of_countability/QY1CRdAy">axiom of countability</a>". Specifically, a space 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is said to be first-countable if each point has a <a href="/facts/Countable/Wj4DmWPv">countable</a> <a href="/facts/Neighbourhood_system/fMRM8Xht">neighbourhood basis</a> (local base). That is, for each point 
  
    
      
        x
      
    
    {\displaystyle x}
  
 in 
  
    
      
        X
      
    
    {\displaystyle X}
  
 there exists a <a href="/facts/Sequence/gV2S4uqp">sequence</a> 
  
    
      
        
          N
          
            1
          
        
        ,
        
          N
          
            2
          
        
        ,
        …
      
    
    {\displaystyle N_{1},N_{2},\ldots }
  
 of <a href="/facts/Neighbourhood_(topology)/XHKgrpkp">neighbourhoods</a> of 
  
    
      
        x
      
    
    {\displaystyle x}
  
 such that for any neighbourhood 
  
    
      
        N
      
    
    {\displaystyle N}
  
 of 
  
    
      
        x
      
    
    {\displaystyle x}
  
 there exists an integer 
  
    
      
        i
      
    
    {\displaystyle i}
  
 with 
  
    
      
        
          N
          
            i
          
        
      
    
    {\displaystyle N_{i}}
  
 <a href="/facts/Subset/SJGERmbA">contained in</a> 
  
    
      
        N
        .
      
    
    {\displaystyle N.}

Since every neighborhood of any point contains an <a href="/facts/Open_set/QfTCIqMu">open</a> neighborhood of that point, the <a href="/facts/Neighbourhood_system/fMRM8Xht">neighbourhood basis</a> can be chosen <a href="/facts/Without_loss_of_generality/gVP4dmEe">without loss of generality</a> to consist of open neighborhoods.
</p>

First-countable space open-in-new

First-countable space