Monotonically normal space

<h2 id="definition">Definition</h2>
<p>A <a href="/facts/Topological_space/v2ut7CbK">topological space</a> 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is called monotonically normal if it satisfies any of the following equivalent definitions:<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a><a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a><a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a><a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>
</p>
<h3>Definition 1</h3>
<p>The space 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is <a href="/facts/T1_space/K4exfuf7">T1</a> and there is a function 
  
    
      
        G
      
    
    {\displaystyle G}
  
 that assigns to each ordered pair 
  
    
      
        (
        A
        ,
        B
        )
      
    
    {\displaystyle (A,B)}
  
 of disjoint closed sets in 
  
    
      
        X
      
    
    {\displaystyle X}
  
 an open set 
  
    
      
        G
        (
        A
        ,
        B
        )
      
    
    {\displaystyle G(A,B)}
  
 such that:
</p>
(i) 
  
    
      
        A
        ⊆
        G
        (
        A
        ,
        B
        )
        ⊆
        
          
            
              G
              (
              A
              ,
              B
              )
            
            ¯
          
        
        ⊆
        X
        ∖
        B
      
    
    {\displaystyle A\subseteq G(A,B)\subseteq {\overline {G(A,B)}}\subseteq X\setminus B}
  
;
(ii) 
  
    
      
        G
        (
        A
        ,
        B
        )
        ⊆
        G
        (
        
          A
          ′
        
        ,
        
          B
          ′
        
        )
      
    
    {\displaystyle G(A,B)\subseteq G(A',B')}
  
 whenever 
  
    
      
        A
        ⊆
        
          A
          ′
        
      
    
    {\displaystyle A\subseteq A'}
  
 and 
  
    
      
        
          B
          ′
        
        ⊆
        B
      
    
    {\displaystyle B'\subseteq B}
  
.
<p>Condition (i) says 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is a normal space, as witnessed by the function 
  
    
      
        G
      
    
    {\displaystyle G}
  
.  
Condition (ii) says that 
  
    
      
        G
        (
        A
        ,
        B
        )
      
    
    {\displaystyle G(A,B)}
  
 varies in a monotone fashion, hence the terminology <i>monotonically normal</i>.
The operator 
  
    
      
        G
      
    
    {\displaystyle G}
  
 is called a monotone normality operator.
</p><p>One can always choose 
  
    
      
        G
      
    
    {\displaystyle G}
  
 to satisfy the property
</p>

G
        (
        A
        ,
        B
        )
        ∩
        G
        (
        B
        ,
        A
        )
        =
        ∅
      
    
    {\displaystyle G(A,B)\cap G(B,A)=\emptyset }
  
,
<p>by replacing each 
  
    
      
        G
        (
        A
        ,
        B
        )
      
    
    {\displaystyle G(A,B)}
  
 by 
  
    
      
        G
        (
        A
        ,
        B
        )
        ∖
        
          
            
              G
              (
              B
              ,
              A
              )
            
            ¯
          
        
      
    
    {\displaystyle G(A,B)\setminus {\overline {G(B,A)}}}
  
.
</p>
<h3>Definition 2</h3>
<p>The space 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is T1 and there is a function 
  
    
      
        G
      
    
    {\displaystyle G}
  
 that assigns to each ordered pair 
  
    
      
        (
        A
        ,
        B
        )
      
    
    {\displaystyle (A,B)}
  
 of <a href="/facts/Separated_sets/NxVgxIF9">separated sets</a> in 
  
    
      
        X
      
    
    {\displaystyle X}
  
 (that is, such that 
  
    
      
        A
        ∩
        
          
            B
            ¯
          
        
        =
        B
        ∩
        
          
            A
            ¯
          
        
        =
        ∅
      
    
    {\displaystyle A\cap {\overline {B}}=B\cap {\overline {A}}=\emptyset }
  
) an open set 
  
    
      
        G
        (
        A
        ,
        B
        )
      
    
    {\displaystyle G(A,B)}
  
 satisfying the same conditions (i) and (ii) of Definition 1.
</p>
<h3>Definition 3</h3>
<p>The space 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is T1 and there is a function 
  
    
      
        μ
      
    
    {\displaystyle \mu }
  
 that assigns to each pair 
  
    
      
        (
        x
        ,
        U
        )
      
    
    {\displaystyle (x,U)}
  
 with 
  
    
      
        U
      
    
    {\displaystyle U}
  
 open in 
  
    
      
        X
      
    
    {\displaystyle X}
  
 and 
  
    
      
        x
        ∈
        U
      
    
    {\displaystyle x\in U}
  
 an open set 
  
    
      
        μ
        (
        x
        ,
        U
        )
      
    
    {\displaystyle \mu (x,U)}
  
 such that:
</p>
(i) 
  
    
      
        x
        ∈
        μ
        (
        x
        ,
        U
        )
      
    
    {\displaystyle x\in \mu (x,U)}
  
;
(ii) if 
  
    
      
        μ
        (
        x
        ,
        U
        )
        ∩
        μ
        (
        y
        ,
        V
        )
        ≠
        ∅
      
    
    {\displaystyle \mu (x,U)\cap \mu (y,V)\neq \emptyset }
  
, then 
  
    
      
        x
        ∈
        V
      
    
    {\displaystyle x\in V}
  
 or 
  
    
      
        y
        ∈
        U
      
    
    {\displaystyle y\in U}
  
.
<p>Such a function 
  
    
      
        μ
      
    
    {\displaystyle \mu }
  
 automatically satisfies
</p>

x
        ∈
        μ
        (
        x
        ,
        U
        )
        ⊆
        
          
            
              μ
              (
              x
              ,
              U
              )
            
            ¯
          
        
        ⊆
        U
      
    
    {\displaystyle x\in \mu (x,U)\subseteq {\overline {\mu (x,U)}}\subseteq U}
  
.
<p>(<i>Reason</i>: Suppose 
  
    
      
        y
        ∈
        X
        ∖
        U
      
    
    {\displaystyle y\in X\setminus U}
  
. Since 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is T1, there is an open neighborhood 
  
    
      
        V
      
    
    {\displaystyle V}
  
 of 
  
    
      
        y
      
    
    {\displaystyle y}
  
 such that 
  
    
      
        x
        ∉
        V
      
    
    {\displaystyle x\notin V}
  
. By condition (ii), 
  
    
      
        μ
        (
        x
        ,
        U
        )
        ∩
        μ
        (
        y
        ,
        V
        )
        =
        ∅
      
    
    {\displaystyle \mu (x,U)\cap \mu (y,V)=\emptyset }
  
, that is, 
  
    
      
        μ
        (
        y
        ,
        V
        )
      
    
    {\displaystyle \mu (y,V)}
  
 is  a neighborhood of 
  
    
      
        y
      
    
    {\displaystyle y}
  
 disjoint from 
  
    
      
        μ
        (
        x
        ,
        U
        )
      
    
    {\displaystyle \mu (x,U)}
  
.  So 
  
    
      
        y
        ∉
        
          
            
              μ
              (
              x
              ,
              U
              )
            
            ¯
          
        
      
    
    {\displaystyle y\notin {\overline {\mu (x,U)}}}
  
.)<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a>
</p>
<h3>Definition 4</h3>
<p>Let 
  
    
      
        
          
            B
          
        
      
    
    {\displaystyle {\mathcal {B}}}
  
 be a <a href="/facts/Base_(topology)/8EDSDVPD">base</a> for the <a href="/facts/Topology_(structure)/v2ut7CbK">topology</a> of 
  
    
      
        X
      
    
    {\displaystyle X}
  
.
The space 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is T1 and there is a function 
  
    
      
        μ
      
    
    {\displaystyle \mu }
  
 that assigns to each pair 
  
    
      
        (
        x
        ,
        U
        )
      
    
    {\displaystyle (x,U)}
  
 with 
  
    
      
        U
        ∈
        
          
            B
          
        
      
    
    {\displaystyle U\in {\mathcal {B}}}
  
 and 
  
    
      
        x
        ∈
        U
      
    
    {\displaystyle x\in U}
  
 an open set 
  
    
      
        μ
        (
        x
        ,
        U
        )
      
    
    {\displaystyle \mu (x,U)}
  
 satisfying the same conditions (i) and (ii) of Definition 3.
</p>
<h3>Definition 5</h3>
<p>The space 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is T1 and there is a function 
  
    
      
        μ
      
    
    {\displaystyle \mu }
  
 that assigns to each pair 
  
    
      
        (
        x
        ,
        U
        )
      
    
    {\displaystyle (x,U)}
  
 with 
  
    
      
        U
      
    
    {\displaystyle U}
  
 open in 
  
    
      
        X
      
    
    {\displaystyle X}
  
 and 
  
    
      
        x
        ∈
        U
      
    
    {\displaystyle x\in U}
  
 an open set 
  
    
      
        μ
        (
        x
        ,
        U
        )
      
    
    {\displaystyle \mu (x,U)}
  
 such that:
</p>
(i) 
  
    
      
        x
        ∈
        μ
        (
        x
        ,
        U
        )
      
    
    {\displaystyle x\in \mu (x,U)}
  
;
(ii) if 
  
    
      
        U
      
    
    {\displaystyle U}
  
 and 
  
    
      
        V
      
    
    {\displaystyle V}
  
 are open and 
  
    
      
        x
        ∈
        U
        ⊆
        V
      
    
    {\displaystyle x\in U\subseteq V}
  
, then 
  
    
      
        μ
        (
        x
        ,
        U
        )
        ⊆
        μ
        (
        x
        ,
        V
        )
      
    
    {\displaystyle \mu (x,U)\subseteq \mu (x,V)}
  
;
(iii) if 
  
    
      
        x
      
    
    {\displaystyle x}
  
 and 
  
    
      
        y
      
    
    {\displaystyle y}
  
 are distinct points, then 
  
    
      
        μ
        (
        x
        ,
        X
        ∖
        {
        y
        }
        )
        ∩
        μ
        (
        y
        ,
        X
        ∖
        {
        x
        }
        )
        =
        ∅
      
    
    {\displaystyle \mu (x,X\setminus \{y\})\cap \mu (y,X\setminus \{x\})=\emptyset }
  
.
<p>Such a function 
  
    
      
        μ
      
    
    {\displaystyle \mu }
  
 automatically satisfies all conditions of Definition 3.
</p>
<h2 id="examples">Examples</h2>
<ul><li>Every <a href="/facts/Metrizable_space/BCLdo5Jf">metrizable space</a> is monotonically normal.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a></li>
<li>Every <a href="/facts/Order_topology/EHISytz6">linearly ordered topological space</a> (LOTS) is monotonically normal.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a><a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a>  This is assuming the <a href="/facts/Axiom_of_Choice/1Ura2HTh">Axiom of Choice</a>, as without it there are examples of LOTS that are not even normal.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a></li>
<li>The <a href="/facts/Sorgenfrey_line/uE1nTKxG">Sorgenfrey line</a> is monotonically normal.<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a>  This follows from Definition 4 by taking as a base for the topology all intervals of the form 
  
    
      
        [
        a
        ,
        b
        )
      
    
    {\displaystyle [a,b)}
  
 and for 
  
    
      
        x
        ∈
        [
        a
        ,
        b
        )
      
    
    {\displaystyle x\in [a,b)}
  
 by letting 
  
    
      
        μ
        (
        x
        ,
        [
        a
        ,
        b
        )
        )
        =
        [
        x
        ,
        b
        )
      
    
    {\displaystyle \mu (x,[a,b))=[x,b)}
  
.  Alternatively, the Sorgenfrey line is monotonically normal because it can be embedded as a subspace of a LOTS, namely the <a href="/facts/Double_arrow_space/d0blXrvg">double arrow space</a>.</li>
<li>Any <a href="/facts/Generalised_metric/0yyMaqW6">generalised metric</a> is monotonically normal.</li></ul>
<h2 id="properties">Properties</h2>
<ul><li>Monotone normality is a <a href="/facts/Hereditary_property/J64xYUGG">hereditary property</a>: Every subspace of a monotonically normal space is monotonically normal.</li>
<li>Every monotonically normal space is <a href="/facts/Completely_normal_Hausdorff_space/t4DLqLCd">completely normal Hausdorff</a> (or T5).</li>
<li>Every monotonically normal space is <a href="/facts/Hereditarily_collectionwise_normal/2gAkshCr">hereditarily collectionwise normal</a>.<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a></li>
<li>The image of a monotonically normal space under a continuous <a href="/facts/Closed_map/abHUKX7l">closed map</a> is monotonically normal.<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a></li>
<li>A compact Hausdorff space 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is the continuous image of a compact linearly ordered space if and only if 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is monotonically normal.<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a><a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a></li></ul>

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

<ol>
<li id="fn:1"><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:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Borges, Carlos R. (March 1973). "A Study of Monotonically Normal Spaces" (PDF). Proceedings of the American Mathematical Society. 38 (1): 211–214. doi:10.2307/2038799. JSTOR 2038799. <a href="https://www.ams.org/proc/1973-038-01/S0002-9939-1973-0324644-4/S0002-9939-1973-0324644-4.pdf" target="_blank">https://www.ams.org/proc/1973-038-01/S0002-9939-1973-0324644-4/S0002-9939-1973-0324644-4.pdf</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Bennett, Harold; Lutzer, David (2015). "Mary Ellen Rudin and monotone normality" (PDF). Topology and Its Applications. 195: 50–62. doi:10.1016/j.topol.2015.09.021. <a href="https://www.sciencedirect.com/science/article/pii/S0166864115003946/pdfft?md5=03a782ebd040aefa11d033e4ebe31e88&pid=1-s2.0-S0166864115003946-main.pdf" target="_blank">https://www.sciencedirect.com/science/article/pii/S0166864115003946/pdfft?md5=03a782ebd040aefa11d033e4ebe31e88&pid=1-s2.0-S0166864115003946-main.pdf</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Brandsma, Henno. "monotone normality, linear orders and the Sorgenfrey line". Ask a Topologist. <a href="http://at.yorku.ca/b/ask-a-topologist/2003/0383.htm" target="_blank">http://at.yorku.ca/b/ask-a-topologist/2003/0383.htm</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Zhang, Hang; Shi, Wei-Xue (2012). "Monotone normality and neighborhood assignments" (PDF). Topology and Its Applications. 159 (3): 603–607. doi:10.1016/j.topol.2011.10.007. <a href="https://www.sciencedirect.com/science/article/pii/S0166864111004664/pdf?md5=fd8e6c9493d1c1097662ece3609d49c3&pid=1-s2.0-S0166864111004664-main.pdf" target="_blank">https://www.sciencedirect.com/science/article/pii/S0166864111004664/pdf?md5=fd8e6c9493d1c1097662ece3609d49c3&pid=1-s2.0-S0166864111004664-main.pdf</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Brandsma, Henno. "monotone normality, linear orders and the Sorgenfrey line". Ask a Topologist. <a href="http://at.yorku.ca/b/ask-a-topologist/2003/0383.htm" target="_blank">http://at.yorku.ca/b/ask-a-topologist/2003/0383.htm</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>Heath, Lutzer, Zenor, Theorem 5.3 <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
<li id="fn:8"><p>Brandsma, Henno. "monotone normality, linear orders and the Sorgenfrey line". Ask a Topologist. <a href="http://at.yorku.ca/b/ask-a-topologist/2003/0383.htm" target="_blank">http://at.yorku.ca/b/ask-a-topologist/2003/0383.htm</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li>
<li id="fn:9"><p>van Douwen, Eric K. (September 1985). "Horrors of Topology Without AC: A Nonnormal Orderable Space" (PDF). Proceedings of the American Mathematical Society. 95 (1): 101–105. doi:10.2307/2045582. JSTOR 2045582. <a href="/wiki/Eric_van_Douwen" target="_blank">/wiki/Eric_van_Douwen</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li>
<li id="fn:10"><p>Brandsma, Henno. "monotone normality, linear orders and the Sorgenfrey line". Ask a Topologist. <a href="http://at.yorku.ca/b/ask-a-topologist/2003/0383.htm" target="_blank">http://at.yorku.ca/b/ask-a-topologist/2003/0383.htm</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li>
<li id="fn:11"><p>Heath, Lutzer, Zenor, Theorem 3.1 <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li>
<li id="fn:12"><p>Heath, Lutzer, Zenor, Theorem 2.6 <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li>
<li id="fn:13"><p>Rudin, Mary Ellen (2001). "Nikiel's conjecture" (PDF). Topology and Its Applications. 116 (3): 305–331. doi:10.1016/S0166-8641(01)00218-8. <a href="https://www.sciencedirect.com/science/article/pii/S0166864101002188/pdf?md5=9558d29000bd32218f70f02c2d63883a&pid=1-s2.0-S0166864101002188-main.pdf" target="_blank">https://www.sciencedirect.com/science/article/pii/S0166864101002188/pdf?md5=9558d29000bd32218f70f02c2d63883a&pid=1-s2.0-S0166864101002188-main.pdf</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li>
<li id="fn:14"><p>Bennett, Harold; Lutzer, David (2015). "Mary Ellen Rudin and monotone normality" (PDF). Topology and Its Applications. 195: 50–62. doi:10.1016/j.topol.2015.09.021. <a href="https://www.sciencedirect.com/science/article/pii/S0166864115003946/pdfft?md5=03a782ebd040aefa11d033e4ebe31e88&pid=1-s2.0-S0166864115003946-main.pdf" target="_blank">https://www.sciencedirect.com/science/article/pii/S0166864115003946/pdfft?md5=03a782ebd040aefa11d033e4ebe31e88&pid=1-s2.0-S0166864115003946-main.pdf</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li>
</ol>

Monotonically normal space open-in-new

Monotonically normal space