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

is T1 such that:

1. ;

2. .

varies in a monotone fashion, hence the terminology monotonically normal is called a monotone normality operator.

to satisfy the property

1. ,

.

Definition 2

is T1 of separated sets satisfying the same conditions (i) and (ii) of Definition 1.

Definition 3

is T1 such that:

1. ;

2. .

automatically satisfies

1. .

(Reason is T1.)

Definition 4

be a base for the topology is T1 satisfying the same conditions (i) and (ii) of Definition 3.

Definition 5

is T1 such that:

1. ;

2. ;

3. .

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.. 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.

• is monotonically normal.

References