Hereditarily normal space: Difference between revisions
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===Stronger properties=== | ===Stronger properties=== | ||
* [[Perfectly normal space]] | * [[Weaker than::Perfectly normal space]] | ||
* [[Monotonically normal space]] | * [[Weaker than::Monotonically normal space]] | ||
* [[Hereditarily collectionwise normal space]] | * [[Weaker than::Hereditarily collectionwise normal space]] | ||
* [[Elastic space]] | * [[Weaker than::Elastic space]] | ||
* [[Metrizable space]] | * [[Weaker than::Metrizable space]] | ||
* [[Linearly orderable space]] | * [[Weaker than::Linearly orderable space]] | ||
* [[CW-space]] | * [[Weaker than::CW-space]] | ||
===Weaker properties=== | ===Weaker properties=== | ||
* [[Normal space]] | * [[Stronger than::Normal space]] | ||
==Metaproperties== | ==Metaproperties== | ||
Revision as of 01:25, 17 January 2009
This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces
This is a variation of normality. View other variations of normality
In the T family (properties of topological spaces related to separation axioms), this is called: T5
Definition
Symbol-free definition
A topological space is said to be hereditarily normal or completely normal (sometimes also totally normal) if it is T1 and satisfies the following equivalent conditions:
- Every subspace of it is normal under the subspace topology
- Given two separated subsets of the topological space (viz two subsets such that neither intersects the closure of the other), there exist disjoint open sets containing them
Relation with other properties
Stronger properties
- Perfectly normal space
- Monotonically normal space
- Hereditarily collectionwise normal space
- Elastic space
- Metrizable space
- Linearly orderable space
- CW-space
Weaker properties
Metaproperties
Hereditariness
This property of topological spaces is hereditary, or subspace-closed. In other words, any subspace (subset with the subspace topology) of a topological space with this property also has this property.
View other subspace-hereditary properties of topological spaces
By the first definition, it is clear that any subspace of a hereditarily normal space is hereditarily normal.
References
Textbook references
- Topology (2nd edition) by James R. MunkresMore info, Page 205, Exercise 6 (definition introduced in exercise): Introduced using term completely normal space