Cocountable topology

<p>The cocountable topology, also known as the countable complement topology, is a <a href="/facts/Topology/2EqW47cU">topology</a> that can be defined on any <a href="/facts/Infinite_set/9rTN0xGH">infinite set</a> 
  
    
      
        X
      
    
    {\displaystyle X}
  
. In this topology, a set is <a href="/facts/Open_set/QfTCIqMu">open</a> if its <a href="/facts/Set_complement/yWUePIYY">complement</a> in 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is either countable or equal to the entire set. Equivalently, the open sets consist of the <a href="/facts/Empty_set/ffEk3eA6">empty set</a> and all subsets of 
  
    
      
        X
      
    
    {\displaystyle X}
  
 whose complements are countable, a property known as <a href="/facts/Cocountable/b1xPg6Yt">cocountability</a>. The only <a href="/facts/Closed_set/upYnRTUj">closed sets</a> in this topology are 
  
    
      
        X
      
    
    {\displaystyle X}
  
 itself and the countable subsets of 
  
    
      
        X
      
    
    {\displaystyle X}
  
.
</p>

Cocountable topology open-in-new

Cocountable topology