Monotonically normal space

{{short description|Property of topological spaces stronger than normality}}

In mathematics, specifically in the field of topology, a '''monotonically normal space''' is a particular kind of normal space, defined in terms of a monotone normality operator. It satisfies some interesting properties; for example metric spaces and linearly ordered spaces are monotonically normal, and every monotonically normal space is hereditarily normal.

==Definition==

A topological space <math>X</math> is called '''monotonically normal''' if it satisfies any of the following equivalent definitions:<ref>{{cite journal |last1=Heath |first1=R. W. |last2=Lutzer |first2=D. J. |last3=Zenor |first3=P. L. |date=April 1973 |title=Monotonically Normal Spaces |journal=Transactions of the American Mathematical Society |volume=178 |pages=481–493 |url=https://www.ams.org/tran/1973-178-00/S0002-9947-1973-0372826-2/S0002-9947-1973-0372826-2.pdf |doi=10.2307/1996713|jstor=1996713 |doi-access=free }}</ref><ref>{{cite journal |last=Borges |first=Carlos R. |date=March 1973 |title=A Study of Monotonically Normal Spaces |journal=Proceedings of the American Mathematical Society |volume=38 |number=1 |pages=211–214 |url=https://www.ams.org/proc/1973-038-01/S0002-9939-1973-0324644-4/S0002-9939-1973-0324644-4.pdf |doi=10.2307/2038799|jstor=2038799 |doi-access=free }}</ref><ref name="Rudin">{{cite journal |last1=Bennett |first1=Harold |last2=Lutzer |first2=David |title=Mary Ellen Rudin and monotone normality |journal=Topology and Its Applications |date=2015 |volume=195 |pages=50–62 |doi=10.1016/j.topol.2015.09.021 |url=https://www.sciencedirect.com/science/article/pii/S0166864115003946/pdfft?md5=03a782ebd040aefa11d033e4ebe31e88&pid=1-s2.0-S0166864115003946-main.pdf}}</ref><ref name="Brandsma">{{cite web |last1=Brandsma |first1=Henno |title=monotone normality, linear orders and the Sorgenfrey line |url=http://at.yorku.ca/b/ask-a-topologist/2003/0383.htm |website=Ask a Topologist}}</ref>

===Definition 1===

The space <math>X</math> is T1 and there is a function <math>G</math> that assigns to each ordered pair <math>(A,B)</math> of disjoint closed sets in <math>X</math> an open set <math>G(A,B)</math> such that: :(i) <math>A\subseteq G(A,B)\subseteq \overline{G(A,B)}\subseteq X\setminus B</math>; :(ii) <math>G(A,B)\subseteq G(A',B')</math> whenever <math>A\subseteq A'</math> and <math>B'\subseteq B</math>.

Condition (i) says <math>X</math> is a normal space, as witnessed by the function <math>G</math>. Condition (ii) says that <math>G(A,B)</math> varies in a monotone fashion, hence the terminology ''monotonically normal''. The operator <math>G</math> is called a '''monotone normality operator'''.

One can always choose <math>G</math> to satisfy the property :<math>G(A,B)\cap G(B,A)=\emptyset</math>, by replacing each <math>G(A,B)</math> by <math>G(A,B)\setminus\overline{G(B,A)}</math>.

===Definition 2===

The space <math>X</math> is T1 and there is a function <math>G</math> that assigns to each ordered pair <math>(A,B)</math> of separated sets in <math>X</math> (that is, such that <math>A\cap\overline{B}=B\cap\overline{A}=\emptyset</math>) an open set <math>G(A,B)</math> satisfying the same conditions (i) and (ii) of Definition 1.

===Definition 3===

Such a function <math>\mu</math> automatically satisfies :<math>x\in\mu(x,U)\subseteq\overline{\mu(x,U)}\subseteq U</math>. (''Reason'': Suppose <math>y\in X\setminus U</math>. Since <math>X</math> is T1, there is an open neighborhood <math>V</math> of <math>y</math> such that <math>x\notin V</math>. By condition (ii), <math>\mu(x,U)\cap\mu(y,V)=\emptyset</math>, that is, <math>\mu(y,V)</math> is a neighborhood of <math>y</math> disjoint from <math>\mu(x,U)</math>. So <math>y\notin\overline{\mu(x,U)}</math>.)<ref>{{cite journal |last1=Zhang |first1=Hang |last2=Shi |first2=Wei-Xue |title=Monotone normality and neighborhood assignments |journal=Topology and Its Applications |date=2012 |volume=159 |issue=3 |pages=603–607 |doi=10.1016/j.topol.2011.10.007 |url=https://www.sciencedirect.com/science/article/pii/S0166864111004664/pdf?md5=fd8e6c9493d1c1097662ece3609d49c3&pid=1-s2.0-S0166864111004664-main.pdf}}</ref>

===Definition 4===

Let <math>\mathcal{B}</math> be a base for the topology of <math>X</math>. The space <math>X</math> is T1 and there is a function <math>\mu</math> that assigns to each pair <math>(x,U)</math> with <math>U\in\mathcal{B}</math> and <math>x\in U</math> an open set <math>\mu(x,U)</math> satisfying the same conditions (i) and (ii) of Definition 3.

===Definition 5===

The space <math>X</math> is T1 and there is a function <math>\mu</math> that assigns to each pair <math>(x,U)</math> with <math>U</math> open in <math>X</math> and <math>x\in U</math> an open set <math>\mu(x,U)</math> such that: :(i) <math>x\in\mu(x,U)</math>; :(ii) if <math>U</math> and <math>V</math> are open and <math>x\in U\subseteq V</math>, then <math>\mu(x,U)\subseteq\mu(x,V)</math>; :(iii) if <math>x</math> and <math>y</math> are distinct points, then <math>\mu(x,X\setminus\{y\})\cap\mu(y,X\setminus\{x\})=\emptyset</math>.

Such a function <math>\mu</math> automatically satisfies all conditions of Definition 3.

==Examples==

* Every metrizable space is monotonically normal.<ref name="Brandsma" /> * Every linearly ordered topological space (LOTS) is monotonically normal.<ref>Heath, Lutzer, Zenor, Theorem 5.3</ref><ref name="Brandsma" /> This is assuming the Axiom of Choice, as without it there are examples of LOTS that are not even normal.<ref>{{cite journal |last=van Douwen |first=Eric K. |authorlink=Eric van Douwen |date=September 1985 |title=Horrors of Topology Without AC: A Nonnormal Orderable Space |journal=Proceedings of the American Mathematical Society |volume=95 |number=1 |pages=101–105 |url=https://www.ams.org/proc/1985-095-01/S0002-9939-1985-0796455-5/S0002-9939-1985-0796455-5.pdf |doi=10.2307/2045582|jstor=2045582 }}</ref> * The Sorgenfrey line is monotonically normal.<ref name="Brandsma" /> This follows from Definition 4 by taking as a base for the topology all intervals of the form <math>[a,b)</math> and for <math>x\in[a,b)</math> by letting <math>\mu(x,[a,b))=[x,b)</math>. Alternatively, the Sorgenfrey line is monotonically normal because it can be embedded as a subspace of a LOTS, namely the double arrow space. * Any generalised metric is monotonically normal.

==Properties==

* Monotone normality is a hereditary property: Every subspace of a monotonically normal space is monotonically normal. * Every monotonically normal space is completely normal Hausdorff (or T5). * Every monotonically normal space is hereditarily collectionwise normal.<ref>Heath, Lutzer, Zenor, Theorem 3.1</ref> * The image of a monotonically normal space under a continuous closed map is monotonically normal.<ref>Heath, Lutzer, Zenor, Theorem 2.6</ref> * A compact Hausdorff space <math>X</math> is the continuous image of a compact linearly ordered space if and only if <math>X</math> is monotonically normal.<ref>{{cite journal |last1=Rudin |first1=Mary Ellen |title=Nikiel's conjecture |journal=Topology and Its Applications |date=2001 |volume=116 |issue=3 |pages=305–331 |doi=10.1016/S0166-8641(01)00218-8 |url=https://www.sciencedirect.com/science/article/pii/S0166864101002188/pdf?md5=9558d29000bd32218f70f02c2d63883a&pid=1-s2.0-S0166864101002188-main.pdf}}</ref><ref name="Rudin" />

==References==

Category:Properties of topological spaces