Hereditarily normal space

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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

Weaker properties



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.


Textbook references

  • Topology (2nd edition) by James R. MunkresMore info, Page 205, Exercise 6 (definition introduced in exercise): Introduced using term completely normal space