Normal space

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This article defines a property of topological space that is pivotal (viz important) among currently studied properties of topological spaces

In the T family (properties of topological spaces related to separation axioms), this is called: T4

For survey articles related to this, refer: Category:Survey articles related to normality

Definition

Symbol-free definition

A topological space is said to be normal if it satisfies the following equivalent conditions:

All points in it are closed sets, and given any two disjoint closed subsets in the topological space, there are disjoint open sets containing them.
All points in it are closed sets, and given any two disjoint closed subsets, there is a continuous function taking the value $0$ at one closed set and 1 at the other
All points are closed, and every point-finite open cover possesses a shrinking.

Some versions of the definition omit the condition that points are closed subsets. This is a different, and weaker, notion, and is covered here as normal-minus-Hausdorff space.

Definition with symbols

A topological space $X$ is said to be normal if it satisfies the following equivalent conditions:

For all $x \in X$ , the set ${x}$ is closed, and for any two closed subsets $A,B \subseteq X$ , there exist open subsets $U,V \subseteq X$ such that $A \subseteq U, B \subseteq V$ , and $U \cap V = \varnothing$ .
For all $x \in X$ , the set ${x}$ is closed, and for any two closed subsets $A,B \subseteq X$ , there exists a continuous map $f:X \to [0,1]$ such that $f(x) = 0 \ \forall x \in A$ and $f(x) = 1 \ \forall \ x \in A$ .
For all $x \in X$ , the set ${x}$ is closed, and 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 $\overline{V_i} \subseteq U_i$ .

Some versions of the definition omit the condition that points are closed subsets. This is a different, and weaker, notion, and is covered here as normal-minus-Hausdorff space.

Equivalence of definitions

The direction (2) implies (1) is easy: if there is a continuous function $f:X \to [0,1]$ such that $A \subseteq f^{-1}(\{ 0 \})$ and $B \subseteq f^{-1}(\{ 1 \})$ , then we can take the open sets $f - 1 ((0,1 / 2))$ and $f - 1 ((1 / 2,1))$ .

The direction (1) implies (2) follows from Urysohn's lemma.

Relation with other properties

This property is a pivotal (important) member of its property space. Its variations, opposites, and other properties related to it and defined using it are often studied

View all variations of normality

Stronger properties

property	quick description	proof of implication	proof of strictness (reverse implication failure)	intermediate notions
Compact Hausdorff space	a compact space that is also Hausdorff	compact Hausdorff implies normal	normal not implies compact	click here
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	click here
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 G-delta subset	perfectly normal implies normal	normal not implies perfectly normal	click here
Metrizable space	can be given the structure of a metric space with the same topology	metrizable implies normal	normal not implies metrizable	click here
CW-space	the underlying topological space of a CW-complex	CW implies normal	normal not implies CW	click here
Linearly orderable space	obtained using the order topology for some linear ordering	linearly orderable implies normal	normal not implies linearly orderable	click here
Collectionwise normal space
Monotonically normal space

Weaker properties

property	quick description	proof of implication	proof of strictness (reverse implication failure)	intermediate notions
Completely regular space (also called $T 3.5$ )	continuous function taking 0 at point, 1 at disjoint closed set	normal implies completely regular	completely regular not implies normal
Regular space (also called $T 3$ )	open subsets separating point and disjoint closed subset	normal implies regular	regular not implies normal	click here
Hausdorff space	open subsets separating distinct points	(via regular)	(via regular)	click here
T1 space	every point is closed	by definition	(via Hausdorff, regular)	click here
Kolmogorov space (also called $T 0$ )	for any two points, open subset containing one and not the other	by definition		click here

Metaproperties

Metaproperty name	Satisfied?	Proof	Section in this article
product-closed property of topological spaces	No	normality is not product-closed	#Products
subspace-hereditary property of topological spaces	No	normality is not hereditary	#Hereditariness
weakly hereditary property of topological spaces	Yes	normality is weakly hereditary	#Weak hereditariness
refining-preserved property of topological spaces	No	normality is not refining-preserved	#Refining

Products

NO: This property of topological spaces is not a product-closed property of topological spaces: a product of topological spaces, each satisfying the property, when equipped with the product topology, does not necessarily satisfy the property.
View other properties that are not product-closed

A product of two normal spaces, endowed with the product topology, need not be normal. For full proof, refer: Normality is not product-closed

Hereditariness

This property of topological spaces is not hereditary on all subsets

It is possible to have a normal space $X$ and a subspace $Y$ of $X$ such that $Y$ is not a normal space. For full proof, refer: Normality is not hereditary

Further information: Complete regularity is hereditary, normal implies completely regular, metrizable implies hereditarily normal, CW implies hereditarily normal

Weak hereditariness

This property of topological spaces is weakly hereditary or closed subspace-closed; in other words, any closed subset (equipped with the subspace topology) of a space with the property, also has the property.
View all weakly hereditary properties of topological spaces | View all subspace-hereditary properties of topological spaces

Any subspace of a normal space need not be normal. However, any closed subset of a normal space is normal, under the subspace topology. Further information: Normality is weakly hereditary

Refining

NO: This property of topological spaces is not a refining-preserved property of topological spaces. In other words, putting a finer topology on a topological space satisfying this property might give a topological space not satisfying this property.
View other properties that are not refining-preserved

Moving to a finer topology, i.e., adding more open subsets, may destroy the property of normality. For full proof, refer: Normality is not refining-preserved

Further information: Hausdorffness is refining-preserved, T1 is refining-preserved, regularity is not refining-preserved, metrizability is not refining-preserved

Facts

Any connected normal space having at least two points (and more generally, any connected Urysohn space having at least two points) is uncountable. For full proof, refer: connected Urysohn implies uncountable

Effect of property operators

The subspace operator

Applying the subspace operator to this property gives: completely regular space

A topological space can be realized as a subspace of a normal space iff it is completely regular. Necessity follows from the fact that normal spaces are completely regular, and any subspace of a completely regular space is completely regular. Sufficiency follows from the Stone-Cech compactification.

The hereditarily operator

Applying the hereditarily operator to this property gives: hereditarily normal space

A topological space in which every subspace is normal is termed hereditarily normal (some people call it completely normal). Note that metrizable spaces are hereditarily normal.

The locally operator

Applying the locally operator to this property gives: locally normal space

References

Textbook references

Topology (2nd edition) by James R. Munkres^{More info}, Page 195,Chapter 4, Section 31 (formal definition, along with definition of regular space)
Lecture Notes on Elementary Topology and Geometry (Undergraduate Texts in Mathematics) by I. M. Singer and J. A. Thorpe^{More info}, Page 28 (formal definition)

External links

Definition links

Wikipedia page

Facts about Normal spaceRDF feed

Defined in	Munkres +, and SingerThorpe +
Dissatisfies metaproperty	Product-closed property of topological spaces +, Subspace-hereditary property of topological spaces +, and Refining-preserved property of topological spaces +
Referenced in	Munkres (?, ?, ?) +, and SingerThorpe (?, ?, ?) +
Satisfies metaproperty	Weakly hereditary property of topological spaces +
Stronger than	Completely regular space +, Regular space +, Hausdorff space +, T1 space +, and Kolmogorov space +
Weaker than	Compact Hausdorff space +, Hereditarily normal space +, Paracompact Hausdorff space +, Regular Lindelof space +, Perfectly normal space +, Metrizable space +, CW-space +, Linearly orderable space +, Collectionwise normal space +, and Monotonically normal space +

Normal space

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