Normal Hausdorff space
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 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 , the set is closed in , and 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 | all points in it are closed sets, and given any two disjoint closed subsets, there is a continuous function taking the value at one closed set and 1 at the other. | For all , the set is closed in , and for any two closed subsets , such that , there exists a continuous map (to the closed unit interval) such that and . |
3 | point-finite open cover has shrinking | all points are closed, and every point-finite open cover possesses a shrinking. | for all , the set is closed in , and for any point-finite open cover of , there exists a shrinking : the form an open cover and . |
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 , the set is closed in , and given any finite collection of pairwise disjoint closed subsets , there exist pairwise disjoint open subsets of such that for all . |
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 such that and , then we can take the open sets and .
The direction (1) implies (2) follows from Urysohn's lemma.
This article is about a basic definition in topology.
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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
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
Weaker properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
Completely regular space (also called ) | 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 ) | 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 ) | for any two points, open subset containing one and not the other | by definition | |FULL LIST, MORE INFO |
Metaproperties
[SHOW MORE]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)