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]\displaystyle{ X }[/math] is called monotonically normal if it satisfies any of the following equivalent definitions:

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].)

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.
• Every linearly ordered topological space (LOTS) is monotonically normal. This is assuming the Axiom of Choice, as without it there are examples of LOTS that are not even normal.
• The Sorgenfrey line is monotonically normal. 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.
• The image of a monotonically normal space under a continuous closed map is monotonically normal.
• 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.