Monotonically normal space

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 is called monotonically normal if it satisfies any of the following equivalent definitions:

Definition 1

The space is T₁ and there is a function that assigns to each ordered pair of disjoint closed sets in an open set such that:
Condition says is a normal space, as witnessed by the function.
Condition says that varies in a monotone fashion, hence the terminology monotonically normal.
The operator is called a monotone normality operator.
One can always choose to satisfy the property
by replacing each by.

Definition 2

The space is T₁ and there is a function that assigns to each ordered pair of separated sets in an open set satisfying the same conditions and of Definition 1.

Definition 3

The space is T₁ and there is a function that assigns to each pair with open in and an open set such that:
Such a function automatically satisfies

Definition 4

Let be a base for the topology of.
The space is T₁ and there is a function that assigns to each pair with and an open set satisfying the same conditions and of Definition 3.

Definition 5

The space is T₁ and there is a function that assigns to each pair with open in and an open set such that:
Such a function automatically satisfies all conditions of Definition 3.

Examples

Every metrizable space is monotonically normal.
Every linearly ordered topological space 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 and for by letting. 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.
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 is the continuous image of a compact linearly ordered space if and only if is monotonically normal.