Hereditarily normal space

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

• Normal space

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