Dense-in-itself

<h2 id="examples">Examples</h2>
<p>A simple example of a set that is dense-in-itself but not closed (and hence not a perfect set) is the set of <a href="/facts/Irrational_numbers/Psv19TET">irrational numbers</a> (considered as a subset of the <a href="/facts/Real_numbers/R02gw5Pb">real numbers</a>). This set is dense-in-itself because every <a href="/facts/Neighbourhood_(mathematics)/XHKgrpkp">neighborhood</a> of an irrational number 
  
    
      
        x
      
    
    {\displaystyle x}
  
 contains at least one other irrational number 
  
    
      
        y
        ≠
        x
      
    
    {\displaystyle y\neq x}
  
. On the other hand, the set of irrationals is not closed because every <a href="/facts/Rational_number/VJQmyY05">rational number</a> lies in its <a href="/facts/Closure_(topology)/9x2wlgUW">closure</a>. Similarly, the set of rational numbers is also dense-in-itself but not closed in the space of real numbers.
</p><p>The above examples, the irrationals and the rationals, are also <a href="/facts/Dense_set/nPT9Yxao">dense sets</a> in their topological space, namely 
  
    
      
        
          R
        
      
    
    {\displaystyle \mathbb {R} }
  
. As an example that is dense-in-itself but not dense in its topological space, consider 
  
    
      
        
          Q
        
        ∩
        [
        0
        ,
        1
        ]
      
    
    {\displaystyle \mathbb {Q} \cap [0,1]}
  
. This set is not dense in 
  
    
      
        
          R
        
      
    
    {\displaystyle \mathbb {R} }
  
 but is dense-in-itself.
</p>
<h2 id="properties">Properties</h2>
<p>A <a href="/facts/Singleton_(mathematics)/LtNVfujT">singleton</a> subset of a space 
  
    
      
        X
      
    
    {\displaystyle X}
  
 can never be dense-in-itself, because its unique point is isolated in it.
</p><p>The dense-in-itself subsets of any space are closed under <a href="/facts/Union_of_sets/KVVXKfWl">unions</a>.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> In a dense-in-itself space, they include all <a href="/facts/Open_set/QfTCIqMu">open sets</a>.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> In a dense-in-itself <a href="/facts/T1_space/K4exfuf7">T1 space</a> they include all <a href="/facts/Dense_set/nPT9Yxao">dense sets</a>.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> However, spaces that are not T1 may have dense subsets that are not dense-in-itself: for example in the dense-in-itself space 
  
    
      
        X
        =
        {
        a
        ,
        b
        }
      
    
    {\displaystyle X=\{a,b\}}
  
 with the <a href="/facts/Indiscrete_topology/uA7RlGth">indiscrete topology</a>, the set 
  
    
      
        A
        =
        {
        a
        }
      
    
    {\displaystyle A=\{a\}}
  
 is dense, but is not dense-in-itself.
</p><p>The closure of any dense-in-itself set is a <a href="/facts/Perfect_set/wKUjfCkM">perfect set</a>.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a>
</p><p>In general, the <a href="/facts/Intersection/hVW6QFWD">intersection</a> of two dense-in-itself sets is not dense-in-itself.  But the intersection of a dense-in-itself set and an open set is dense-in-itself.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Nowhere_dense_set/m0oF4X8R">Nowhere dense set</a></li>
<li><a href="/facts/Glossary_of_topology/DVkS2W7J">Glossary of topology</a></li>
<li><a href="/facts/Dense_order/EVVQI3V6">Dense order</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li><a href="/facts/Ryszard_Engelking/aAjuMWvC">Engelking, Ryszard</a> (1989). <i>General Topology</i>. Heldermann Verlag, Berlin. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-88538-006-4.</li>
<li><a href="/facts/Kuratowski/8xoKvo5b">Kuratowski, K.</a> (1966). <i>Topology Vol. I</i>. Academic Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 012429202X.</li>
<li><a href="/facts/Lynn_Arthur_Steen/JFdvAfpc">Steen, Lynn Arthur</a>; <a href="/facts/J._Arthur_Seebach%2c_Jr./sU4TbeUA">Seebach, J. Arthur Jr.</a> (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>
<p><i>This article incorporates material from Dense in-itself on <a href="/facts/PlanetMath/wphPCtLf">PlanetMath</a>, which is licensed under the Creative Commons Attribution/Share-Alike License.</i>
</p>

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

<ol>
<li id="fn:1"><p>Steen & Seebach, p. 6 <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Engelking, p. 25 <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Levy, Ronnie; Porter, Jack (1996). "On Two questions of Arhangel'skii and Collins regarding submaximal spaces" (PDF). Topology Proceedings. 21: 143–154. <a href="http://topology.nipissingu.ca/tp/reprints/v21/tp21008.pdf" target="_blank">http://topology.nipissingu.ca/tp/reprints/v21/tp21008.pdf</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Dontchev, Julian; Ganster, Maximilian; Rose, David (1977). "α-Scattered spaces II". <a href="https://www.researchgate.net/publication/228597275" target="_blank">https://www.researchgate.net/publication/228597275</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Engelking, 1.7.10, p. 59 <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Kuratowski, p. 78 <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>Kuratowski, p. 78 <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
<li id="fn:8"><p>Kuratowski, p. 77 <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li>
</ol>

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Dense-in-itself