First-countable space

<h2 id="examples-and-counterexamples">Examples and counterexamples</h2>
<p>The majority of 'everyday' spaces in <a href="/facts/Mathematics/pxTouaz4">mathematics</a> are first-countable. In particular, every <a href="/facts/Metric_space/gnBBRXtc">metric space</a> is first-countable. To see this, note that the set of <a href="/facts/Open_ball/f9vpMVMp">open balls</a> centered at 
  
    
      
        x
      
    
    {\displaystyle x}
  
 with radius 
  
    
      
        1
        
          /
        
        n
      
    
    {\displaystyle 1/n}
  
 for integers  form a countable local base at 
  
    
      
        x
        .
      
    
    {\displaystyle x.}

</p><p>An example of a space that is not first-countable is the <a href="/facts/Cofinite_topology/LvpB23pu">cofinite topology</a> on an uncountable set (such as the <a href="/facts/Real_line/6eq42xX5">real line</a>). More generally, the <a href="/facts/Zariski_topology/uolkKBpM">Zariski topology</a> on an <a href="/facts/Algebraic_variety/Vk7f97vn">algebraic variety</a> over an uncountable field is not first-countable.
</p><p>Another counterexample is the <a href="/facts/Ordinal_space/EHISytz6">ordinal space</a> 
  
    
      
        
          ω
          
            1
          
        
        +
        1
        =
        
          [
          
            0
            ,
            
              ω
              
                1
              
            
          
          ]
        
      
    
    {\displaystyle \omega _{1}+1=\left[0,\omega _{1}\right]}
  
 where 
  
    
      
        
          ω
          
            1
          
        
      
    
    {\displaystyle \omega _{1}}
  
 is the <a href="/facts/First_uncountable_ordinal/Quu6AMEy">first uncountable ordinal</a> number. The element 
  
    
      
        
          ω
          
            1
          
        
      
    
    {\displaystyle \omega _{1}}
  
 is a <a href="/facts/Limit_point/95MykoGU">limit point</a> of the subset 
  
    
      
        
          [
          
            0
            ,
            
              ω
              
                1
              
            
          
          )
        
      
    
    {\displaystyle \left[0,\omega _{1}\right)}
  
 even though no sequence of elements in 
  
    
      
        
          [
          
            0
            ,
            
              ω
              
                1
              
            
          
          )
        
      
    
    {\displaystyle \left[0,\omega _{1}\right)}
  
 has the element 
  
    
      
        
          ω
          
            1
          
        
      
    
    {\displaystyle \omega _{1}}
  
 as its limit. In particular, the point 
  
    
      
        
          ω
          
            1
          
        
      
    
    {\displaystyle \omega _{1}}
  
 in the space 
  
    
      
        
          ω
          
            1
          
        
        +
        1
        =
        
          [
          
            0
            ,
            
              ω
              
                1
              
            
          
          ]
        
      
    
    {\displaystyle \omega _{1}+1=\left[0,\omega _{1}\right]}
  
 does not have a countable local base. Since 
  
    
      
        
          ω
          
            1
          
        
      
    
    {\displaystyle \omega _{1}}
  
 is the only such point, however, the subspace 
  
    
      
        
          ω
          
            1
          
        
        =
        
          [
          
            0
            ,
            
              ω
              
                1
              
            
          
          )
        
      
    
    {\displaystyle \omega _{1}=\left[0,\omega _{1}\right)}
  
 is first-countable.
</p><p>The <a href="/facts/Quotient_space_(topology)/muE0qQVC">quotient space</a> 
  
    
      
        
          R
        
        
          /
        
        
          N
        
      
    
    {\displaystyle \mathbb {R} /\mathbb {N} }
  
 where the natural numbers on the real line are identified as a single point is not first countable.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> However, this space has the property that for any subset 
  
    
      
        A
      
    
    {\displaystyle A}
  
 and every element 
  
    
      
        x
      
    
    {\displaystyle x}
  
 in the closure of 
  
    
      
        A
        ,
      
    
    {\displaystyle A,}
  
 there is a sequence in 
  
    
      
        A
      
    
    {\displaystyle A}
  
 converging to 
  
    
      
        x
        .
      
    
    {\displaystyle x.}
  
 A space with this sequence property is sometimes called a <a href="/facts/Fr%C3%A9chet%E2%80%93Urysohn_space/ccN00rSh">Fréchet–Urysohn space</a>.
</p><p>First-countability is strictly weaker than <a href="/facts/Second-countability/qw9YdnBN">second-countability</a>. Every <a href="/facts/Second-countable_space/qw9YdnBN">second-countable space</a> is first-countable, but any uncountable <a href="/facts/Discrete_space/hpvXA23u">discrete space</a> is first-countable but not second-countable.
</p>
<h2 id="properties">Properties</h2>
<p>One of the most important properties of first-countable spaces is that given a subset 
  
    
      
        A
        ,
      
    
    {\displaystyle A,}
  
 a point 
  
    
      
        x
      
    
    {\displaystyle x}
  
 lies in the <a href="/facts/Closure_(topology)/9x2wlgUW">closure</a> of 
  
    
      
        A
      
    
    {\displaystyle A}
  
 if and only if there exists a <a href="/facts/Sequence/gV2S4uqp">sequence</a> 
  
    
      
        
          
            (
            
              x
              
                n
              
            
            )
          
          
            n
            =
            1
          
          
            ∞
          
        
      
    
    {\displaystyle \left(x_{n}\right)_{n=1}^{\infty }}
  
 in 
  
    
      
        A
      
    
    {\displaystyle A}
  
 that <a href="/facts/Limit_of_a_sequence/Ktge2RwL">converges</a> to 
  
    
      
        x
        .
      
    
    {\displaystyle x.}
  
 (In other words, every first-countable space is a <a href="/facts/Fr%C3%A9chet-Urysohn_space/ccN00rSh">Fréchet-Urysohn space</a> and thus also a <a href="/facts/Sequential_space/xAjFpdfR">sequential space</a>.) This has consequences for <a href="/facts/Limit_of_a_function/HFbQ69e5">limits</a> and <a href="/facts/Continuity_(topology)/sKbl02pB">continuity</a>. In particular, if 
  
    
      
        f
      
    
    {\displaystyle f}
  
 is a function on a first-countable space, then 
  
    
      
        f
      
    
    {\displaystyle f}
  
 has a limit 
  
    
      
        L
      
    
    {\displaystyle L}
  
 at the point 
  
    
      
        x
      
    
    {\displaystyle x}
  
 if and only if for every sequence 
  
    
      
        
          x
          
            n
          
        
        →
        x
        ,
      
    
    {\displaystyle x_{n}\to x,}
  
 where 
  
    
      
        
          x
          
            n
          
        
        ≠
        x
      
    
    {\displaystyle x_{n}\neq x}
  
 for all 
  
    
      
        n
        ,
      
    
    {\displaystyle n,}
  
 we have 
  
    
      
        f
        
          (
          
            x
            
              n
            
          
          )
        
        →
        L
        .
      
    
    {\displaystyle f\left(x_{n}\right)\to L.}
  
 Also, if 
  
    
      
        f
      
    
    {\displaystyle f}
  
 is a function on a first-countable space, then 
  
    
      
        f
      
    
    {\displaystyle f}
  
 is continuous if and only if whenever 
  
    
      
        
          x
          
            n
          
        
        →
        x
        ,
      
    
    {\displaystyle x_{n}\to x,}
  
 then 
  
    
      
        f
        
          (
          
            x
            
              n
            
          
          )
        
        →
        f
        (
        x
        )
        .
      
    
    {\displaystyle f\left(x_{n}\right)\to f(x).}

</p><p>In first-countable spaces, <a href="/facts/Sequentially_compact_space/5vL96D9l">sequential compactness</a> and <a href="/facts/Countably_compact_space/Ppm9uXq5">countable compactness</a> are equivalent properties. However, there exist examples of sequentially compact, first-countable spaces that are not compact (these are necessarily not metrizable spaces).  One such space is the <a href="/facts/Order_topology/EHISytz6">ordinal space</a> 
  
    
      
        
          [
          
            0
            ,
            
              ω
              
                1
              
            
          
          )
        
        .
      
    
    {\displaystyle \left[0,\omega _{1}\right).}
  
 Every first-countable space is <a href="/facts/Compactly_generated_space/2A5umYGa">compactly generated</a>.
</p><p>Every <a href="/facts/Subspace_(topology)/NjU6z4Ca">subspace</a> of a first-countable space is first-countable. Any countable <a href="/facts/Product_space/0hg02IaX">product</a> of a first-countable space is first-countable, although uncountable products need not be.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Fr%C3%A9chet%E2%80%93Urysohn_space/ccN00rSh">Fréchet–Urysohn space</a> – Property of topological space</li>
<li><a href="/facts/Second-countable_space/qw9YdnBN">Second-countable space</a> – Topological space whose topology has a countable base</li>
<li><a href="/facts/Separable_space/2bmU73bk">Separable space</a> – Topological space with a dense countable subset</li>
<li><a href="/facts/Sequential_space/xAjFpdfR">Sequential space</a> – Topological space characterized by sequences</li></ul>

<h2 id="bibliography">Bibliography</h2>
<ul><li><a href="https://www.encyclopediaofmath.org/index.php?title=first_axiom_of_countability">"first axiom of countability"</a>, <i><a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a></i>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a>, 2001 [1994]</li>
<li><a href="/facts/Ryszard_Engelking/aAjuMWvC">Engelking, Ryszard</a> (1989). <i>General Topology</i>. Sigma Series in Pure Mathematics, Vol. 6 (Revised and completed ed.). Heldermann Verlag, Berlin. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3885380064.</li></ul>

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

<ol>
<li id="fn:1"><p>(Engelking 1989, Example 1.6.18) - Engelking, Ryszard (1989). General Topology. Sigma Series in Pure Mathematics, Vol. 6 (Revised and completed ed.). Heldermann Verlag, Berlin. ISBN 3885380064. <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
</ol>

First-countable space open-in-new

First-countable space