Normal Hausdorff space

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There are two alternative definitions of the term. Please see: Convention:Hausdorffness assumption

Definition

Equivalent definitions in tabular format

No. Shorthand A topological space is said to be normal Hausdorff if ... A topological space X is said to be normal Hausdorff if ...
1 separation of disjoint closed subsets by open subsets 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. for all x \in X, the set \{ x \} is closed in X, and 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 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. For all x \in X, the set \{ x \} is closed in X, and 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 all points are closed, and every point-finite open cover possesses a shrinking. for all x \in X, the set \{ x \} is closed in X, 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.
4 separation of finitely many disjoint closed subsets by open subsets all points in it are closed sets, and given a finite collection of pairwise disjoint closed subsets in the topological space, there are pairwise disjoint open sets containing them. for all x \in X, the set \{ x \} is closed in X, and given any finite collection of pairwise disjoint closed subsets A_1,A_2,\dots,A_n \subseteq X, there exist pairwise disjoint open subsets U_1,U_2, \dots, U_n of X such that A_i \subseteq U_i for all i.

Note that in each of the definitions, the T1 space assumption (that points are closed) can be replaced by the (a priori stronger) Hausdorff space assumption, without changing the meaning of the overall definition.

Some people use the term normal space for what is called here a normal Hausdorff space; however, we define the term normal space as not having the T1 space assumption.

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.


This article is about a basic definition in topology.
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View a complete list of basic definitions in topology

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

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

Stronger properties

Property Meaning 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 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

Weaker properties

Property Meaning 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 |FULL LIST, MORE INFO
Regular space (also called T_3) open subsets separating point and disjoint closed subset normal implies regular regular not implies normal Completely regular space|FULL LIST, MORE INFO
Hausdorff space open subsets separating distinct points (via regular) (via regular) Completely regular space|FULL LIST, MORE INFO
T1 space every point is closed by definition (via Hausdorff, regular) Completely regular space|FULL LIST, MORE INFO
Kolmogorov space (also called T_0) for any two points, open subset containing one and not the other by definition |FULL LIST, MORE INFO

Metaproperties

Metaproperty name Satisfied? Proof
product-closed property of topological spaces No normality is not product-closed
subspace-hereditary property of topological spaces No normality is not hereditary
weakly hereditary property of topological spaces Yes normality is weakly hereditary
refining-preserved property of topological spaces No normality is not refining-preserved
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Facts

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. MunkresMore 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. ThorpeMore info, Page 28 (formal definition)

External links

Definition links