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:
 For any disjoint closed subsets , contains and its closure is disjoint from
 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 spaceView 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 spaceFULL LIST, MORE INFO

ordered fieldmetrizable space 
underlying topology of a space with a metric taking values in an ordered field 
ordered fieldmetrizable implies monotonically normal 
monotonically normal not implies ordered fieldmetrizable 
FULL LIST, MORE INFO

linearly orderable space 
order topology from a linear ordering on a set 
linearly orderable implies monotonically normal 
monotonically normal not implies linearly orderable 
FULL LIST, MORE INFO

elastic space 

elastic implies monotonically normal 
monotonically normal not implies elastic 
FULL LIST, MORE INFO

closed subEuclidean space 

(via metrizable) 
(via metrizable) 
Elastic space, Metrizable space, Protometrizable spaceFULL LIST, MORE INFO

manifold 

(via metrizable) 
(via metrizable) 
Elastic space, Metrizable space, Protometrizable spaceFULL 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 spaceFULL 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 spaceFULL 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 spaceFULL 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 
FULL LIST, MORE INFO

completely regular space 

(via normal) 
(via normal) 
Normal Hausdorff spaceFULL LIST, MORE INFO

regular space 

(via normal) 
(via normal) 
Normal Hausdorff spaceFULL LIST, MORE INFO

Hausdorff space 

(via normal) 
(via normal) 
Normal Hausdorff spaceFULL LIST, MORE INFO

Urysohn space 

(via normal) 
(via normal) 
FULL LIST, MORE INFO

collectionwise Hausdorff space 

(via collectionwise normal) 
(via collectionwise normal) 
Collectionwise normal space, Hereditarily collectionwise normal spaceFULL LIST, MORE INFO

Incomparable properties
Metaproperties
Hereditariness
This property of topological spaces is hereditary, or subspaceclosed. In other words, any subspace (subset with the subspace topology) of a topological space with this property also has this property.
View other subspacehereditary properties of topological spaces
Any subspace of a monotonically normal space is monotonically normal. For full proof, refer: Monotone normality is hereditary