Please see Convention:Hausdorffness assumption
Definition
Equivalent definitions in tabular format
No. 
Shorthand 
A topological space is said to be normal(minusHausdorff) if ... 
A topological space is said to be normal(minusHausdorff) if ...

1 
separation of disjoint closed subsets by open subsets 
given any two disjoint closed subsets in the topological space, there are disjoint open sets containing them. 
given any two closed subsets such that , there exist disjoint open subsets of such that , and .

2 
separation of disjoint closed subsets by continuous functions 
given any two disjoint closed subsets, there is a continuous function taking the value at one closed set and 1 at the other. 
for any two closed subsets , such that , there exists a continuous map (to the closed unit interval) such that and .

3 
pointfinite open cover has shrinking 
every pointfinite open cover possesses a shrinking. 
for any pointfinite open cover of , there exists a shrinking : the form an open cover and .

Relation with other properties
Stronger properties
Property 
Meaning 
Proof of implication 
Proof of strictness (reverse implication failure) 
Intermediate notions

normal Hausdorff space 
a normal space that is also Hausdorff. Some people include Hausdorffness as part of the definition of normal space. 
(obvious) 
a set of size more than one with the trivial topology is normal but not Hausdorff 


compact Hausdorff space 
a compact space that is also Hausdorff 
compact Hausdorff implies normal 
normal not implies compact 
Binormal space, Paracompact Hausdorff spaceFULL LIST, MORE INFO

hereditarily normal space 
every subspace is a normal space under the subspace topology 

normality is not hereditary 


paracompact Hausdorff space 
a paracompact space that is also Hausdorff 
paracompact Hausdorff implies normal 
normal not implies paracompact 
Binormal spaceFULL LIST, MORE INFO

regular Lindelof space 
both a regular space and a Lindelof space 
regular Lindelof implies normal 
normal not implies Lindelof 


perfectly normal space 
every closed subset is a Gdelta subset 
perfectly normal implies normal 
normal not implies perfectly normal 
Hereditarily normal spaceFULL LIST, MORE INFO

metrizable space 
can be given the structure of a metric space with the same topology 
metrizable implies normal 
normal not implies metrizable 
Collectionwise normal space, Elastic space, Hereditarily collectionwise normal space, Hereditarily normal space, Monotonically normal space, Paracompact Hausdorff space, Perfectly normal space, Protometrizable spaceFULL LIST, MORE INFO

CWspace 
the underlying topological space of a CWcomplex 
CW implies normal 
normal not implies CW 
Hereditarily normal space, Paracompact Hausdorff space, Perfectly normal spaceFULL LIST, MORE INFO

linearly orderable space 
obtained using the order topology for some linear ordering 
linearly orderable implies normal 
normal not implies linearly orderable 
Collectionwise normal space, Hereditarily collectionwise normal space, Hereditarily normal space, Monotonically normal spaceFULL LIST, MORE INFO

collectionwise normal space 




monotonically normal space 




ultraconnected space 

ultraconnected implies normal 


