Hereditarily normal space
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Contents
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. Munkres^{More info}, Page 205, Exercise 6 (definition introduced in exercise): Introduced using term completely normal space