Hereditarily normal space: Difference between revisions
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===Symbol-free definition=== | ===Symbol-free definition=== | ||
A [[topological space]] is said to be '''hereditarily normal''' if it is [[T1 space|T1]] and satisfies the following equivalent conditions: | A [[topological space]] is said to be '''hereditarily normal''' or '''completely normal''' (sometimes also '''totally normal''') if it is [[T1 space|T1]] and 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]] | ||
Revision as of 20:49, 17 December 2007
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
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.