Collectionwise normal space

<h2 id="properties">Properties</h2>
<ul><li>A collectionwise normal space is <a href="/facts/Collectionwise_Hausdorff/RUo1kuXa">collectionwise Hausdorff</a>.</li>
<li>A collectionwise normal space is <a href="/facts/Normal_space/t4DLqLCd">normal</a>.</li>
<li>A <a href="/facts/Hausdorff_space/gbKIrusl">Hausdorff</a> <a href="/facts/Paracompact_space/rjTrvF7j">paracompact space</a> is collectionwise normal.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>  In particular, every <a href="/facts/Metrizable_space/BCLdo5Jf">metrizable space</a> is collectionwise normal.Note: The Hausdorff condition is necessary here, since for example an infinite set with the <a href="/facts/Cofinite_topology/LvpB23pu">cofinite topology</a> is <a href="/facts/Compact_space/d0cgXJH7">compact</a>, hence paracompact, and T1, but is not even normal.</li>
<li>Every normal <a href="/facts/Countably_compact_space/Ppm9uXq5">countably compact space</a> (hence every normal compact space) is collectionwise normal.<i>Proof</i>: Use the fact that in a countably compact space any discrete family of nonempty subsets is finite.</li>
<li>An <a href="/facts/F-sigma/vI6Y6fdQ">Fσ</a>-set in a collectionwise normal space is also collectionwise normal in the <a href="/facts/Subspace_topology/NjU6z4Ca">subspace topology</a>. In particular, this holds for closed subsets.</li>
<li>The <i>Moore metrization theorem</i> states that a collectionwise normal <a href="/facts/Moore_space_(topology)/AwgvhlqJ">Moore space</a> is <a href="/facts/Metrizable/BCLdo5Jf">metrizable</a>.</li></ul>
<h2 id="hereditarily-collectionwise-normal-space">Hereditarily collectionwise normal space</h2>
<p>A topological space <i>X</i> is called hereditarily collectionwise normal if every subspace of <i>X</i> with the subspace topology is collectionwise normal.
</p><p>In the same way that <a href="/facts/Hereditarily_normal_space/t4DLqLCd">hereditarily normal spaces</a> can be characterized in terms of <a href="/facts/Separated_sets/NxVgxIF9">separated sets</a>, there is an equivalent characterization for hereditarily collectionwise normal spaces.  A family 
  
    
      
        
          F
          
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        (
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    {\displaystyle F_{i}(i\in I)}
  
 of subsets of <i>X</i> is called a separated family if for every <i>i</i>, we have 
  
    
      
        
          F
          
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        ∩
        cl
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          ⋃
          
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    {\textstyle F_{i}\cap \operatorname {cl} (\bigcup _{j\neq i}F_{j})=\emptyset }
  
, with cl denoting the <a href="/facts/Closure_(topology)/9x2wlgUW">closure</a> operator in <i>X</i>, in other words if the family of 
  
    
      
        
          F
          
            i
          
        
      
    
    {\displaystyle F_{i}}
  
 is discrete in its union.  The following conditions are equivalent:<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>
</p>
<ol><li><i>X</i> is hereditarily collectionwise normal.</li>
<li>Every open subspace of <i>X</i> is collectionwise normal.</li>
<li>For every separated family 
  
    
      
        
          F
          
            i
          
        
      
    
    {\displaystyle F_{i}}
  
 of subsets of <i>X</i>, there exists a pairwise disjoint family of open sets 
  
    
      
        
          U
          
            i
          
        
        (
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    {\displaystyle U_{i}(i\in I)}
  
, such that 
  
    
      
        
          F
          
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        ⊆
        
          U
          
            i
          
        
      
    
    {\displaystyle F_{i}\subseteq U_{i}}
  
.</li></ol>
<h3>Examples of hereditarily collectionwise normal spaces</h3>
<ul><li>Every <a href="/facts/Order_topology/EHISytz6">linearly ordered topological space</a> (LOTS)<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a><a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a></li>
<li>Every <a href="/facts/Order_topology/EHISytz6">generalized ordered space</a> (GO-space)</li>
<li>Every <a href="/facts/Metrizable_space/BCLdo5Jf">metrizable space</a>.  This follows from the fact that metrizable spaces are collectionwise normal and being metrizable is a hereditary property.</li>
<li>Every <a href="/facts/Monotonically_normal_space/6lXPWIsw">monotonically normal space</a><a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></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></ul>

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

<ol>
<li id="fn:1"><p>Engelking, Theorem 5.1.17, shows the equivalence between the two definitions (under the assumption of T1, but the proof does not use the T1 property). <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Engelking 1989, Theorem 5.1.18. - Engelking, Ryszard (1989). General Topology. Heldermann Verlag, Berlin. ISBN 3-88538-006-4. <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Engelking 1989, Problem 5.5.1. - Engelking, Ryszard (1989). General Topology. Heldermann Verlag, Berlin. ISBN 3-88538-006-4. <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Steen, Lynn A. (1970). "A direct proof that a linearly ordered space is hereditarily collectionwise normal". Proc. Amer. Math. Soc. 24: 727–728. doi:10.1090/S0002-9939-1970-0257985-7. <a href="/wiki/Lynn_A._Steen" target="_blank">/wiki/Lynn_A._Steen</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Cater, Frank S. (2006). "A Simple Proof that a Linearly Ordered Space is Hereditarily and Completely Collectionwise Normal". Rocky Mountain Journal of Mathematics. 36 (4): 1149–1151. doi:10.1216/rmjm/1181069408. ISSN 0035-7596. JSTOR 44239306. Zbl 1134.54317. <a href="https://projecteuclid.org/journals/rocky-mountain-journal-of-mathematics/volume-36/issue-4/A-Simple-Proof-that-a-Linearly-Ordered-Space-is-Hereditarily/10.1216/rmjm/1181069408.full" target="_blank">https://projecteuclid.org/journals/rocky-mountain-journal-of-mathematics/volume-36/issue-4/A-Simple-Proof-that-a-Linearly-Ordered-Space-is-Hereditarily/10.1216/rmjm/1181069408.full</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Heath, R. W.; Lutzer, D. J.; Zenor, P. L. (April 1973). "Monotonically Normal Spaces" (PDF). Transactions of the American Mathematical Society. 178: 481–493. doi:10.2307/1996713. JSTOR 1996713. <a href="https://www.ams.org/tran/1973-178-00/S0002-9947-1973-0372826-2/S0002-9947-1973-0372826-2.pdf" target="_blank">https://www.ams.org/tran/1973-178-00/S0002-9947-1973-0372826-2/S0002-9947-1973-0372826-2.pdf</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
</ol>

Collectionwise normal space open-in-new

Collectionwise normal space