First uncountable ordinal

<h2 id="topological-properties">Topological properties</h2>
<p>Any ordinal number can be turned into a <a href="/facts/Topological_space/v2ut7CbK">topological space</a> by using the <a href="/facts/Order_topology/EHISytz6">order topology</a>. When viewed as a topological space, 
  
    
      
        
          ω
          
            1
          
        
      
    
    {\displaystyle \omega _{1}}
  
 is often written as 
  
    
      
        [
        0
        ,
        
          ω
          
            1
          
        
        )
      
    
    {\displaystyle [0,\omega _{1})}
  
, to emphasize that it is the space consisting of all ordinals smaller than 
  
    
      
        
          ω
          
            1
          
        
      
    
    {\displaystyle \omega _{1}}
  
.
</p><p>If the <a href="/facts/Axiom_of_countable_choice/goSPxEdj">axiom of countable choice</a> holds, every <a href="/facts/Sequence/gV2S4uqp">increasing ω-sequence</a> of elements of 
  
    
      
        [
        0
        ,
        
          ω
          
            1
          
        
        )
      
    
    {\displaystyle [0,\omega _{1})}
  
 converges to a <a href="/facts/Limit_of_a_sequence/Ktge2RwL">limit</a> in 
  
    
      
        [
        0
        ,
        
          ω
          
            1
          
        
        )
      
    
    {\displaystyle [0,\omega _{1})}
  
. The reason is that the <a href="/facts/Union_(set_theory)/KVVXKfWl">union</a> (i.e., supremum) of every countable set of countable ordinals is another countable ordinal.
</p><p>The topological space 
  
    
      
        [
        0
        ,
        
          ω
          
            1
          
        
        )
      
    
    {\displaystyle [0,\omega _{1})}
  
 is <a href="/facts/Sequentially_compact/5vL96D9l">sequentially compact</a>, but not <a href="/facts/Compact_space/d0cgXJH7">compact</a>. As a consequence, it is not <a href="/facts/Metrizable_space/BCLdo5Jf">metrizable</a>. It is, however, <a href="/facts/Countably_compact_space/Ppm9uXq5">countably compact</a> and thus not <a href="/facts/Lindel%25C3%25B6f_space/HC4t6bvz">Lindelöf</a> (a countably compact space is compact if and only if it is Lindelöf). In terms of <a href="/facts/Axioms_of_countability/QY1CRdAy">axioms of countability</a>, 
  
    
      
        [
        0
        ,
        
          ω
          
            1
          
        
        )
      
    
    {\displaystyle [0,\omega _{1})}
  
 is <a href="/facts/First-countable_space/HDuTYa6A">first-countable</a>, but neither <a href="/facts/Separable_space/2bmU73bk">separable</a> nor <a href="/facts/Second-countable_space/qw9YdnBN">second-countable</a>. 
</p><p>The space 
  
    
      
        [
        0
        ,
        
          ω
          
            1
          
        
        ]
        =
        
          ω
          
            1
          
        
        +
        1
      
    
    {\displaystyle [0,\omega _{1}]=\omega _{1}+1}
  
 is compact and not first-countable. 
  
    
      
        
          ω
          
            1
          
        
      
    
    {\displaystyle \omega _{1}}
  
 is used to define the <a href="/facts/Long_line_(topology)/bgkMDrpB">long line</a> and the <a href="/facts/Tychonoff_plank/o7FR4s91">Tychonoff plank</a>—two important counterexamples in <a href="/facts/Topology/2EqW47cU">topology</a>.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Epsilon_numbers_(mathematics)/t7w8KgA9">Epsilon numbers (mathematics)</a></li>
<li><a href="/facts/Large_countable_ordinal/u0Ea19nd">Large countable ordinal</a></li>
<li><a href="/facts/Ordinal_arithmetic/satc9FHy">Ordinal arithmetic</a></li></ul>

<h2 id="bibliography">Bibliography</h2>

<ul><li>Thomas Jech, <i>Set Theory</i>, 3rd millennium ed., 2003, Springer Monographs in Mathematics, Springer, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-44085-2.</li>
<li>Lynn Arthur Steen and J. Arthur Seebach, Jr., <i><a href="/facts/Counterexamples_in_Topology/f3eAqwaP">Counterexamples in Topology</a></i>. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-486-68735-X (Dover edition).</li></ul>

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

<ol>
<li id="fn:1"><p>"Set Theory > Basic Set Theory (Stanford Encyclopedia of Philosophy)". plato.stanford.edu. Retrieved 2020-08-12. <a href="https://plato.stanford.edu/entries/set-theory/basic-set-theory.html" target="_blank">https://plato.stanford.edu/entries/set-theory/basic-set-theory.html</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>"first uncountable ordinal in nLab". ncatlab.org. Retrieved 2020-08-12. <a href="https://ncatlab.org/nlab/show/first+uncountable+ordinal" target="_blank">https://ncatlab.org/nlab/show/first+uncountable+ordinal</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
</ol>

First uncountable ordinal open-in-new

First uncountable ordinal