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:

(i) ;

(ii) whenever and .

Condition (i) says is a normal space, as witnessed by the function . Condition (ii) 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 (that is, such that ) an open set satisfying the same conditions (i) and (ii) 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:

(i) ;

(ii) if , then or .

Such a function automatically satisfies

.

(Reason: Suppose . Since is T₁, there is an open neighborhood of such that . By condition (ii), , that is, is a neighborhood of disjoint from . So .)

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 (i) and (ii) 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:

(i) ;

(ii) if and are open and , then ;

(iii) if and are distinct points, then .

Such a function 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 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 (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 is the continuous image of a compact linearly ordered space if and only if is monotonically normal.

References