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Monotonically normal space

Property of topological spaces stronger than normality

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Source updated February 10, 2023

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.

01Definition

A topological space $X$ is called monotonically normal if it satisfies any of the following equivalent definitions:

Definition 1

The space $X$ is T₁ and there is a function $G$ that assigns to each ordered pair $(A,B)$ of disjoint closed sets in $X$ an open set $G(A,B)$ such that:

(i) $A\subseteq G(A,B)\subseteq {\overline {G(A,B)}}\subseteq X\setminus B$ ;

(ii) $G(A,B)\subseteq G(A',B')$ whenever $A\subseteq A'$ and $B'\subseteq B$ .

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

One can always choose $G$ to satisfy the property

$G(A,B)\cap G(B,A)=\emptyset$ ,

by replacing each $G(A,B)$ by $G(A,B)\setminus {\overline {G(B,A)}}$ .

Definition 2

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

Definition 3

The space $X$ is T₁ and there is a function $\mu$ that assigns to each pair $(x,U)$ with $U$ open in $X$ and $x\in U$ an open set $\mu (x,U)$ such that:

(i) $x\in \mu (x,U)$ ;

(ii) if $\mu (x,U)\cap \mu (y,V)\neq \emptyset$ , then $x\in V$ or $y\in U$ .

Such a function $\mu$ automatically satisfies

$x\in \mu (x,U)\subseteq {\overline {\mu (x,U)}}\subseteq U$ .

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

Definition 4

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

Definition 5

The space $X$ is T₁ and there is a function $\mu$ that assigns to each pair $(x,U)$ with $U$ open in $X$ and $x\in U$ an open set $\mu (x,U)$ such that:

(i) $x\in \mu (x,U)$ ;

(ii) if $U$ and $V$ are open and $x\in U\subseteq V$ , then $\mu (x,U)\subseteq \mu (x,V)$ ;

(iii) if $x$ and $y$ are distinct points, then $\mu (x,X\setminus \{y\})\cap \mu (y,X\setminus \{x\})=\emptyset$ .

Such a function $\mu$ automatically satisfies all conditions of Definition 3.

02Examples

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 $[a,b)$ and for $x\in [a,b)$ by letting $\mu (x,[a,b))=[x,b)$ . Alternatively, the Sorgenfrey line is monotonically normal because it can be embedded as a subspace of a LOTS, namely the double arrow space.
Any generalised metric is monotonically normal.

03Properties

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 $X$ is the continuous image of a compact linearly ordered space if and only if $X$ is monotonically normal.

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Sources and credits

This article is adapted from the Wikipedia article “Monotonically normal space”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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