Cocountable topology

<h2 id="definitions">Definitions</h2>
<p>Let 
  
    
      
        X
      
    
    {\displaystyle X}
  
 be an <a href="/facts/Infinite_set/9rTN0xGH">infinite set</a> and let 
  
    
      
        
          
            T
          
        
      
    
    {\displaystyle {\mathcal {T}}}
  
 be the set of <a href="/facts/Subsets/SJGERmbA">subsets</a> of 
  
    
      
        X
      
    
    {\displaystyle X}
  
 such that 
  
    
      
        H
        ∈
        
          
            T
          
        
        
        ⟺
        
        X
        ∖
        H
        
          
             is countable, or
          
        
        
        H
        =
        ∅
      
    
    {\displaystyle H\in {\mathcal {T}}\iff X\setminus H{\mbox{ is countable, or}}\,H=\varnothing }
  
 then 
  
    
      
        
          
            T
          
        
      
    
    {\displaystyle {\mathcal {T}}}
  
 is the countable complement toplogy on 
  
    
      
        X
      
    
    {\displaystyle X}
  
, and the <a href="/facts/Topological_space/v2ut7CbK">topological space</a> 
  
    
      
        T
        =
        (
        X
        ,
        
          
            T
          
        
        )
      
    
    {\displaystyle T=(X,{\mathcal {T}})}
  
 is a countable complement space.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a>
</p><p>Symbolically, the topology is typically written as

T
          
        
        =
        {
        H
        ⊆
        X
        :
        H
        =
        ∅
        
          
             or 
          
        
        X
        ∖
        H
        
          
             is countable
          
        
        }
        .
      
    
    {\displaystyle {\mathcal {T}}=\{H\subseteq X:H=\varnothing {\mbox{ or }}X\setminus H{\mbox{ is countable}}\}.}

</p>
<h3>Double pointed cocountable topology</h3>
<p>Let 
  
    
      
        X
      
    
    {\displaystyle X}
  
 be an <a href="/facts/Uncountable_set/zl2WqyJQ">uncountable set</a>. We define the topology 
  
    
      
        
          
            T
          
        
      
    
    {\displaystyle {\mathcal {T}}}
  
 as all open sets whose complements are countable, along with 
  
    
      
        ∅
      
    
    {\displaystyle \varnothing }
  
 and 
  
    
      
        X
      
    
    {\displaystyle X}
  
 itself.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>
</p>
<h3>Cocountable extension topology</h3>
<p>Let 
  
    
      
        X
      
    
    {\displaystyle X}
  
 be the real line. Now let 
  
    
      
        
          
            
              T
            
          
          
            1
          
        
      
    
    {\displaystyle {\mathcal {T}}_{1}}
  
  be the <a href="/facts/Euclidean_topology/gCcjWJvA">Euclidean topology</a> and 
  
    
      
        
          
            
              T
            
          
          
            2
          
        
      
    
    {\displaystyle {\mathcal {T}}_{2}}
  
 be the cocountable topology on 
  
    
      
        X
      
    
    {\displaystyle X}
  
. The <i>cocountable extension topology</i> is the smallest topology generated by 
  
    
      
        
          
            
              T
            
          
          
            1
          
        
        ∪
        
          
            
              T
            
          
          
            2
          
        
      
    
    {\displaystyle {\mathcal {T}}_{1}\cup {\mathcal {T}}_{2}}
  
.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>
</p>
<h2 id="proof-that-cocountable-topology-is-a-topology">Proof that cocountable topology is a topology</h2>
<p>By definition, the <a href="/facts/Empty_set/ffEk3eA6">empty set</a> 
  
    
      
        ∅
      
    
    {\displaystyle \varnothing }
  
 is an element of 
  
    
      
        
          
            T
          
        
      
    
    {\displaystyle {\mathcal {T}}}
  
. Similarly, the entire set 
  
    
      
        X
        ∈
        
          
            T
          
        
      
    
    {\displaystyle X\in {\mathcal {T}}}
  
, since the <a href="/facts/Set_complement/yWUePIYY">complement</a> of 
  
    
      
        X
      
    
    {\displaystyle X}
  
 relative to itself is the empty set, which is <a href="/facts/Vacuously/QBsue3pP">vacuously</a> countable.
</p><p>Suppose 
  
    
      
        A
        ,
        B
        ∈
        
          
            T
          
        
      
    
    {\displaystyle A,B\in {\mathcal {T}}}
  
. Let 
  
    
      
        H
        =
        A
        ∩
        B
      
    
    {\displaystyle H=A\cap B}
  
. Then
</p><p>
  
    
      
        X
        ∖
        H
        =
        X
        ∖
        (
        A
        ∩
        B
        )
        =
        (
        X
        ∖
        A
        )
        ∪
        (
        X
        ∖
        B
        )
      
    
    {\displaystyle X\setminus H=X\setminus (A\cap B)=(X\setminus A)\cup (X\setminus B)}

</p><p>by <a href="/facts/De_Morgan%2527s_laws/dj3F8xpt"> De Morgan's laws</a>. Since 
  
    
      
        A
        ,
        B
        ∈
        
          
            T
          
        
      
    
    {\displaystyle A,B\in {\mathcal {T}}}
  
, it follows that 
  
    
      
        X
        ∖
        A
      
    
    {\displaystyle X\setminus A}
  
 and 
  
    
      
        X
        ∖
        B
      
    
    {\displaystyle X\setminus B}
  
 are both countable. Because the countable union of countable sets is countable, 
  
    
      
        X
        ∖
        H
      
    
    {\displaystyle X\setminus H}
  
 is also countable. Therefore, 
  
    
      
        H
        =
        A
        ∩
        B
        ∈
        
          
            T
          
        
      
    
    {\displaystyle H=A\cap B\in {\mathcal {T}}}
  
, as its complement is countable.
</p><p>Now let 
  
    
      
        
          
            U
          
        
        ⊆
        
          
            T
          
        
      
    
    {\displaystyle {\mathcal {U}}\subseteq {\mathcal {T}}}
  
. Then
</p><p>
  
    
      
        X
        ∖
        
          (
          
            ⋃
            
              
                U
              
            
          
          )
        
        =
        
          ⋂
          
            U
            ∈
            
              
                U
              
            
          
        
        (
        X
        ∖
        U
        )
      
    
    {\displaystyle X\setminus \left(\bigcup {\mathcal {U}}\right)=\bigcap _{U\in {\mathcal {U}}}(X\setminus U)}

</p><p>again by De Morgan's laws. For each 
  
    
      
        U
        ∈
        
          
            U
          
        
      
    
    {\displaystyle U\in {\mathcal {U}}}
  
, 
  
    
      
        X
        ∖
        U
      
    
    {\displaystyle X\setminus U}
  
 is countable. The countable intersection of countable sets is also countable (assuming 
  
    
      
        
          
            U
          
        
      
    
    {\displaystyle {\mathcal {U}}}
  
 is countable), so 
  
    
      
        S
        ∖
        
          (
          
            ⋃
            
              
                U
              
            
          
          )
        
      
    
    {\displaystyle S\setminus \left(\bigcup {\mathcal {U}}\right)}
  
 is countable. Thus, 
  
    
      
        ⋃
        
          
            U
          
        
        ∈
        
          
            T
          
        
      
    
    {\displaystyle \bigcup {\mathcal {U}}\in {\mathcal {T}}}
  
.
</p><p>Since all three <a href="/facts/Topological_space/v2ut7CbK">open set axioms</a> are met, 
  
    
      
        
          
            T
          
        
      
    
    {\displaystyle {\mathcal {T}}}
  
 is a topology on 
  
    
      
        X
      
    
    {\displaystyle X}
  
.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>
</p>
<h2 id="properties">Properties</h2>
<p>Every set 
  
    
      
        X
      
    
    {\displaystyle X}
  
 with the cocountable topology is <a href="/facts/Lindel%25C3%25B6f_space/HC4t6bvz">Lindelöf</a>, since every nonempty <a href="/facts/Open_set/QfTCIqMu">open set</a> omits only countably many points of 
  
    
      
        X
      
    
    {\displaystyle X}
  
. It is also <a href="/facts/T1_space/K4exfuf7">T1</a>, as all singletons are closed. 
</p><p>If 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is an uncountable set, then any two nonempty open sets <a href="/facts/Set_intersection/RPfUNJh9">intersect</a>, hence, the space is not <a href="/facts/Hausdorff_space/gbKIrusl">Hausdorff</a>. However, in the cocountable topology all convergent sequences are eventually constant, so limits are unique. Since <a href="/facts/Compact_space/d0cgXJH7">compact sets</a> in 
  
    
      
        X
      
    
    {\displaystyle X}
  
 are finite subsets, all compact subsets are closed, another condition usually related to the Hausdorff separation axiom.
</p><p>The cocountable topology on a countable set is the <a href="/facts/Discrete_topology/hpvXA23u">discrete topology</a>. The cocountable topology on an uncountable set is <a href="/facts/Hyperconnected_space/Yu2eFIon">hyperconnected</a>, thus <a href="/facts/Connected_space/5CUTxfoA">connected</a>, <a href="/facts/Locally_connected_space/9qhzYyT7">locally connected</a> and <a href="/facts/Pseudocompact_space/Itcq221R">pseudocompact</a>, but neither <a href="/facts/Limit_point_compact/tB8tbHJV">weakly countably compact</a> nor <a href="/facts/Metacompact_space/QYxP3vsj">countably metacompact</a>, hence not compact.
</p>
<h2 id="examples">Examples</h2>
<ul><li>Uncountable set: On any uncountable set, such as the real numbers 
  
    
      
        
          R
        
      
    
    {\displaystyle \mathbb {R} }
  
, the cocountable topology is a proper subset of the <a href="/facts/Standard_topology/tEtvw03F">standard topology</a>. In this case, the topology is T1 but not Hausdorff, first-countable, nor <a href="/facts/Metrizable/BCLdo5Jf">metrizable</a>.</li></ul>
<ul><li>Countable set: If 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is countable, then every subset of 
  
    
      
        X
      
    
    {\displaystyle X}
  
 has a countable complement. In this case, the cocountable topology is just the <a href="/facts/Discrete_topology/hpvXA23u">discrete topology</a>.</li></ul>
<ul><li>Finite sets: On a finite set, the cocountable topology reduces to the <a href="/facts/Indiscrete_topology/uA7RlGth">indiscrete topology</a>, consisting only of the empty set and the whole set. This is because any proper subset of a finite set has a finite (and hence not countable) complement, violating the openness condition.</li></ul>
<ul><li>Subspace topology: If 
  
    
      
        Y
        ⊆
        X
      
    
    {\displaystyle Y\subseteq X}
  
 and 
  
    
      
        X
      
    
    {\displaystyle X}
  
 carries the cocountable topology, then 
  
    
      
        Y
      
    
    {\displaystyle Y}
  
 inherits the <a href="/facts/Subspace_topology/NjU6z4Ca">subspace topology</a>. This topology on 
  
    
      
        Y
      
    
    {\displaystyle Y}
  
 consists of the empty set,  all of 
  
    
      
        Y
      
    
    {\displaystyle Y}
  
, and all subsets 
  
    
      
        U
        ⊆
        Y
      
    
    {\displaystyle U\subseteq Y}
  
 such that 
  
    
      
        Y
        ∖
        U
      
    
    {\displaystyle Y\setminus U}
  
 is countable.</li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Cofinite_topology/LvpB23pu">Cofinite topology</a></li>
<li><a href="/facts/List_of_topologies/mb4VoHSD">List of topologies</a></li></ul>

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

<ol>
<li id="fn:1"><p>Munkres, James Raymond (2000). Topology (2nd ed.). Upper Saddle River (N. J.): Prentice Hall. ISBN 0-13-181629-2. <a href="0-13-181629-2" target="_blank">0-13-181629-2</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Steen, Lynn Arthur; Seebach, Jr., J. Arthur (1978). "2". Counterexamples in Topology (2nd ed.). New York, NY: Springer New York. p. 50. ISBN 978-1-4612-6290-9. <a href="978-1-4612-6290-9" target="_blank">978-1-4612-6290-9</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Steen, Lynn Arthur; Seebach, Jr., J. Arthur (1978). "2". Counterexamples in Topology (2nd ed.). New York, NY: Springer New York. p. 85. ISBN 978-1-4612-6290-9. <a href="978-1-4612-6290-9" target="_blank">978-1-4612-6290-9</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Steen, Lynn Arthur; Seebach, J. Arthur (1978). "20". Counterexamples in Topology (2nd ed.). New York, NY: Springer New York. ISBN 978-0-387-90312-5. <a href="978-0-387-90312-5" target="_blank">978-0-387-90312-5</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
</ol>

Cocountable topology open-in-new

Cocountable topology