Normal space

Please see Convention:Hausdorffness assumption

Definition

Equivalent definitions in tabular format

No.	Shorthand	A topological space is said to be normal(-minus-Hausdorff) if ...	A topological space $X$ is said to be normal(-minus-Hausdorff) 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 $A,B\subseteq X$ such that $A\cap B=\varnothing$, there exist disjoint open subsets $U,V$ of $X$ such that $A\subseteq U,B\subseteq V$, and $U\cap V=\varnothing$.
2	separation of disjoint closed subsets by continuous functions	given any two disjoint closed subsets, there is a continuous function taking the value $0$ at one closed set and 1 at the other.	for any two closed subsets $A,B\subseteq X$, such that $A\cap B=\varnothing$, there exists a continuous map $f:X\to [0,1]$ (to the closed unit interval) such that $f(x)=0\ \forall x\in A$ and $f(x)=1\ \forall \ x\in B$.
3	point-finite open cover has shrinking	every point-finite open cover possesses a shrinking.	for any point-finite open cover $U_{i},i\in I$ of $X$, there exists a shrinking $V_{i},i\in I$: the $V_{i}$ form an open cover and ${\overline {V_{i}}}\subseteq U_{i}$.

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	|FULL LIST, MORE INFO
compact Hausdorff space	a compact space that is also Hausdorff	compact Hausdorff implies normal	normal not implies compact	Binormal space, Paracompact Hausdorff space|FULL LIST, MORE INFO
hereditarily normal space	every subspace is a normal space under the subspace topology		normality is not hereditary	|FULL LIST, MORE INFO
paracompact Hausdorff space	a paracompact space that is also Hausdorff	paracompact Hausdorff implies normal	normal not implies paracompact	Binormal space|FULL LIST, MORE INFO
regular Lindelof space	both a regular space and a Lindelof space	regular Lindelof implies normal	normal not implies Lindelof	|FULL LIST, MORE INFO
perfectly normal space	every closed subset is a G-delta subset	perfectly normal implies normal	normal not implies perfectly normal	Hereditarily normal space|FULL 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 space|FULL LIST, MORE INFO
CW-space	the underlying topological space of a CW-complex	CW implies normal	normal not implies CW	Hereditarily normal space, Paracompact Hausdorff space, Perfectly normal space|FULL 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 space|FULL LIST, MORE INFO
collectionwise normal space
monotonically normal space
ultraconnected space		ultraconnected implies normal		|FULL LIST, MORE INFO