Hereditarily normal space: Difference between revisions
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==Definition== | ==Definition== | ||
===Symbol-free definition=== | ===Symbol-free definition=== | ||
A [[topological space]] is said to be '''hereditarily normal''' or '''completely normal''' (sometimes also '''totally normal''') if it | A [[topological space]] is said to be '''hereditarily normal''' or '''completely normal''' (sometimes also '''totally normal''') if it satisfies the following equivalent conditions: | ||
* Every subspace of it is [[normal space|normal]] under the [[subspace topology]] | * Every subspace of it is [[normal space|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 | * 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 | ||
{{topospace property}} | |||
{{T family|T5}} | |||
==Relation with other properties== | ==Relation with other properties== | ||
| Line 18: | Line 16: | ||
===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== | ||
Latest revision as of 21:54, 27 January 2012
Definition
Symbol-free definition
A topological space is said to be hereditarily normal or completely normal (sometimes also totally normal) if it 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
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
In the T family (properties of topological spaces related to separation axioms), this is called: T5
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