# Monotonically normal space

## Definition

### Definition with symbols

A topological space  is termed monotonically normal if it is a T1 space (i.e., all points are closed) and there exists an operator  from ordered pairs of disjoint closed sets to open sets, such that:

1. For any disjoint closed subsets ,  contains  and its closure is disjoint from 
2. If  and  with all four sets being closed,  disjoint from , and  disjoint from , we have:



This is the monotonicity condition. Such an operator  is termed a monotone normality operator.

This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces
This is a variation of normality. View other variations of normality

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
metrizable space underlying topology of a metric space metrizable implies monotonically normal monotonically normal not implies metrizable Elastic space, Protometrizable space|FULL LIST, MORE INFO
ordered field-metrizable space underlying topology of a space with a metric taking values in an ordered field ordered field-metrizable implies monotonically normal monotonically normal not implies ordered field-metrizable |
linearly orderable space order topology from a linear ordering on a set linearly orderable implies monotonically normal monotonically normal not implies linearly orderable |
elastic space elastic implies monotonically normal monotonically normal not implies elastic |
closed sub-Euclidean space (via metrizable) (via metrizable) Elastic space, Metrizable space, Protometrizable space|FULL LIST, MORE INFO
manifold (via metrizable) (via metrizable) Elastic space, Metrizable space, Protometrizable space|FULL LIST, MORE INFO

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
normal space any two disjoint closed subsets are separated by disjoint open subsets monotonically normal implies normal normal not implies monotonically normal Collectionwise normal space, Hereditarily collectionwise normal space, Hereditarily normal space|FULL LIST, MORE INFO
hereditarily normal space every subspace is a normal space monotonically normal implies hereditarily normal hereditarily normal implies monotonically normal Hereditarily collectionwise normal space|FULL LIST, MORE INFO
collectionwise normal space every discrete collection of closed subsets can be separated by disjoint open subsets monotonically normal not implies collectionwise normal collectionwise normal not implies monotonically normal Hereditarily collectionwise normal space|FULL LIST, MORE INFO
hereditarily collectionwise normal space every subspace is collectionwise normal monotonically normal implies hereditarily collectionwise normal hereditarily collectionwise normal not implies monotonically normal |
completely regular space (via normal) (via normal) Normal Hausdorff space|FULL LIST, MORE INFO
regular space (via normal) (via normal) Normal Hausdorff space|FULL LIST, MORE INFO
Hausdorff space (via normal) (via normal) Normal Hausdorff space|FULL LIST, MORE INFO
Urysohn space (via normal) (via normal) |
collectionwise Hausdorff space (via collectionwise normal) (via collectionwise normal) Collectionwise normal space, Hereditarily collectionwise normal space|FULL LIST, MORE INFO

## Metaproperties

### Hereditariness

This property of topological spaces is hereditary, or subspace-closed. In other words, any subspace (subset with the subspace topology) of a topological space with this property also has this property.
View other subspace-hereditary properties of topological spaces

Any subspace of a monotonically normal space is monotonically normal. For full proof, refer: Monotone normality is hereditary