Excluded point topology

<h2 id="properties">Properties</h2>
<p>Let 
  
    
      
        X
      
    
    {\displaystyle X}
  
 be a space with the excluded point topology with special point 
  
    
      
        p
        .
      
    
    {\displaystyle p.}

</p><p>The space is <a href="/facts/Compact_(topology)/d0cgXJH7">compact</a>, as the only neighborhood of 
  
    
      
        p
      
    
    {\displaystyle p}
  
 is the whole space.
</p><p>The topology is an <a href="/facts/Alexandrov_topology/GSXq6t17">Alexandrov topology</a>.  The smallest neighborhood of 
  
    
      
        p
      
    
    {\displaystyle p}
  
 is the whole space 
  
    
      
        X
        ;
      
    
    {\displaystyle X;}
  
 the smallest neighborhood of a point 
  
    
      
        x
        ≠
        p
      
    
    {\displaystyle x\neq p}
  
 is the singleton 
  
    
      
        {
        x
        }
        .
      
    
    {\displaystyle \{x\}.}
  
  These smallest neighborhoods are compact.  Their closures are respectively 
  
    
      
        X
      
    
    {\displaystyle X}
  
 and 
  
    
      
        {
        x
        ,
        p
        }
        ,
      
    
    {\displaystyle \{x,p\},}
  
 which are also compact.  So the space is <a href="/facts/Locally_relatively_compact/hesalT7q">locally relatively compact</a> (each point admits a <a href="/facts/Local_base/fMRM8Xht">local base</a> of relatively compact neighborhoods) and <a href="/facts/Locally_compact/hesalT7q">locally compact</a> in the sense that each point has a local base of compact neighborhoods.  But points 
  
    
      
        x
        ≠
        p
      
    
    {\displaystyle x\neq p}
  
 do not admit a local base of closed compact neighborhoods.
</p><p>The space is <a href="/facts/Ultraconnected/THksr5sH">ultraconnected</a>, as any nonempty closed set contains the point 
  
    
      
        p
        .
      
    
    {\displaystyle p.}
  
  Therefore the space is also <a href="/facts/Connected_(topology)/5CUTxfoA">connected</a> and <a href="/facts/Path-connected/5CUTxfoA">path-connected</a>.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Finite_topological_space/HKD3e0qp">Finite topological space</a></li>
<li><a href="/facts/Fort_space/qoXquVHu">Fort space</a></li>
<li><a href="/facts/List_of_topologies/mb4VoHSD">List of topologies</a></li>
<li><a href="/facts/Particular_point_topology/6jGkUzjF">Particular point topology</a></li></ul>

<ul><li><a href="/facts/Lynn_Arthur_Steen/JFdvAfpc">Steen, Lynn Arthur</a>; <a href="/facts/J._Arthur_Seebach,_Jr./sU4TbeUA">Seebach, J. Arthur Jr.</a> (1995) [1978], <i><a href="/facts/Counterexamples_in_Topology/f3eAqwaP">Counterexamples in Topology</a></i> (<a href="/facts/Dover_Publications/RwjvlSFv">Dover</a> reprint of 1978 ed.), Berlin, New York: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-486-68735-3, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0507446">0507446</a></li></ul>

Excluded point topology open-in-new

Excluded point topology