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.

## Contents

* 1 Definition
* 1.1 Definition 1
* 1.2 Definition 2
* 1.3 Definition 3
* 1.4 Definition 4
* 1.5 Definition 5
* 2 Examples
* 3 Properties
* 4 References

## Definition

A topological space [math]\displaystyle{ X }[/math] is called monotonically normal if it satisfies any of the following equivalent definitions:[1][2][3][4]

### Definition 1

The space [math]\displaystyle{ X }[/math] is T1 and there is a function [math]\displaystyle{ G }[/math] that assigns to each ordered pair [math]\displaystyle{ (A,B) }[/math] of disjoint closed sets in [math]\displaystyle{ X }[/math] an open set [math]\displaystyle{ G(A,B) }[/math] such that:

(i) [math]\displaystyle{ A\subseteq G(A,B)\subseteq \overline{G(A,B)}\subseteq X\setminus B }[/math];
(ii) [math]\displaystyle{ G(A,B)\subseteq G(A',B') }[/math] whenever [math]\displaystyle{ A\subseteq A' }[/math] and [math]\displaystyle{ B'\subseteq B }[/math].

Condition (i) says [math]\displaystyle{ X }[/math] is a normal space, as witnessed by the function [math]\displaystyle{ G }[/math]. Condition (ii) says that [math]\displaystyle{ G(A,B) }[/math] varies in a monotone fashion, hence the terminology monotonically normal. The operator [math]\displaystyle{ G }[/math] is called a monotone normality operator.

One can always choose [math]\displaystyle{ G }[/math] to satisfy the property

[math]\displaystyle{ G(A,B)\cap G(B,A)=\emptyset }[/math],

by replacing each [math]\displaystyle{ G(A,B) }[/math] by [math]\displaystyle{ G(A,B)\setminus\overline{G(B,A)} }[/math].

### Definition 2

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

### Definition 3

The space [math]\displaystyle{ X }[/math] is T1 and there is a function [math]\displaystyle{ \mu }[/math] that assigns to each pair [math]\displaystyle{ (x,U) }[/math] with [math]\displaystyle{ U }[/math] open in [math]\displaystyle{ X }[/math] and [math]\displaystyle{ x\in U }[/math] an open set [math]\displaystyle{ \mu(x,U) }[/math] such that:

(i) [math]\displaystyle{ x\in\mu(x,U) }[/math];
(ii) if [math]\displaystyle{ \mu(x,U)\cap\mu(y,V)\ne\emptyset }[/math], then [math]\displaystyle{ x\in V }[/math] or [math]\displaystyle{ y\in U }[/math].

Such a function [math]\displaystyle{ \mu }[/math] automatically satisfies

[math]\displaystyle{ x\in\mu(x,U)\subseteq\overline{\mu(x,U)}\subseteq U }[/math].

(Reason: Suppose [math]\displaystyle{ y\in X\setminus U }[/math]. Since [math]\displaystyle{ X }[/math] is T1, there is an open neighborhood [math]\displaystyle{ V }[/math] of [math]\displaystyle{ y }[/math] such that [math]\displaystyle{ x\notin V }[/math]. By condition (ii), [math]\displaystyle{ \mu(x,U)\cap\mu(y,V)=\emptyset }[/math], that is, [math]\displaystyle{ \mu(y,V) }[/math] is a neighborhood of [math]\displaystyle{ y }[/math] disjoint from [math]\displaystyle{ \mu(x,U) }[/math]. So [math]\displaystyle{ y\notin\overline{\mu(x,U)} }[/math].)[5]

### Definition 4

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

### Definition 5

The space [math]\displaystyle{ X }[/math] is T1 and there is a function [math]\displaystyle{ \mu }[/math] that assigns to each pair [math]\displaystyle{ (x,U) }[/math] with [math]\displaystyle{ U }[/math] open in [math]\displaystyle{ X }[/math] and [math]\displaystyle{ x\in U }[/math] an open set [math]\displaystyle{ \mu(x,U) }[/math] such that:

(i) [math]\displaystyle{ x\in\mu(x,U) }[/math];
(ii) if [math]\displaystyle{ U }[/math] and [math]\displaystyle{ V }[/math] are open and [math]\displaystyle{ x\in U\subseteq V }[/math], then [math]\displaystyle{ \mu(x,U)\subseteq\mu(x,V) }[/math];
(iii) if [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math] are distinct points, then [math]\displaystyle{ \mu(x,X\setminus\\{y\\})\cap\mu(y,X\setminus\\{x\\})=\emptyset }[/math].

Such a function [math]\displaystyle{ \mu }[/math] automatically satisfies all conditions of Definition 3.

## Examples

* Every metrizable space is monotonically normal.[4]
* Every linearly ordered topological space (LOTS) is monotonically normal.[6][4] This is assuming the Axiom of Choice, as without it there are examples of LOTS that are not even normal.[7]
* The Sorgenfrey line is monotonically normal.[4] This follows from Definition 4 by taking as a base for the topology all intervals of the form [math]\displaystyle{ [a,b) }[/math] and for [math]\displaystyle{ x\in[a,b) }[/math] by letting [math]\displaystyle{ \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.[8]
* The image of a monotonically normal space under a continuous closed map is monotonically normal.[9]
* A compact Hausdorff space [math]\displaystyle{ X }[/math] is the continuous image of a compact linearly ordered space if and only if [math]\displaystyle{ X }[/math] is monotonically normal.[10][3]

## References

1. ↑ Heath, R. W.; Lutzer, D. J.; Zenor, P. L. (April 1973). "Monotonically Normal Spaces". Transactions of the American Mathematical Society 178: 481–493. doi:10.2307/1996713. https://www.ams.org/tran/1973-178-00/S0002-9947-1973-0372826-2/S0002-9947-1973-0372826-2.pdf.
2. ↑ Borges, Carlos R. (March 1973). "A Study of Monotonically Normal Spaces". Proceedings of the American Mathematical Society 38 (1): 211–214. doi:10.2307/2038799. https://www.ams.org/proc/1973-038-01/S0002-9939-1973-0324644-4/S0002-9939-1973-0324644-4.pdf.
3. ↑ 3.0 3.1 Bennett, Harold; Lutzer, David (2015). "Mary Ellen Rudin and monotone normality". Topology and Its Applications 195: 50–62. doi:10.1016/j.topol.2015.09.021. https://www.sciencedirect.com/science/article/pii/S0166864115003946/pdfft?md5=03a782ebd040aefa11d033e4ebe31e88&pid=1-s2.0-S0166864115003946-main.pdf.
4. ↑ 4.0 4.1 4.2 4.3 Brandsma, Henno. "monotone normality, linear orders and the Sorgenfrey line". http://at.yorku.ca/b/ask-a-topologist/2003/0383.htm.
5. ↑ Zhang, Hang; Shi, Wei-Xue (2012). "Monotone normality and neighborhood assignments". Topology and Its Applications 159 (3): 603–607. doi:10.1016/j.topol.2011.10.007. https://www.sciencedirect.com/science/article/pii/S0166864111004664/pdf?md5=fd8e6c9493d1c1097662ece3609d49c3&pid=1-s2.0-S0166864111004664-main.pdf.
6. ↑ Heath, Lutzer, Zenor, Theorem 5.3
7. ↑ van Douwen, Eric K. (September 1985). "Horrors of Topology Without AC: A Nonnormal Orderable Space". Proceedings of the American Mathematical Society 95 (1): 101–105. doi:10.2307/2045582. https://www.ams.org/proc/1985-095-01/S0002-9939-1985-0796455-5/S0002-9939-1985-0796455-5.pdf.
8. ↑ Heath, Lutzer, Zenor, Theorem 3.1
9. ↑ Heath, Lutzer, Zenor, Theorem 2.6
10. ↑ Rudin, Mary Ellen (2001). "Nikiel's conjecture". Topology and Its Applications 116 (3): 305–331. doi:10.1016/S0166-8641(01)00218-8. https://www.sciencedirect.com/science/article/pii/S0166864101002188/pdf?md5=9558d29000bd32218f70f02c2d63883a&pid=1-s2.0-S0166864101002188-main.pdf.

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