Hereditarily normal space

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

Stronger properties

 * Weaker than::Perfectly normal space
 * Weaker than::Monotonically normal space
 * Weaker than::Hereditarily collectionwise normal space
 * Weaker than::Elastic space
 * Weaker than::Metrizable space
 * Weaker than::Linearly orderable space
 * Weaker than::CW-space

Weaker properties

 * Stronger than::Normal space

Metaproperties
By the first definition, it is clear that any subspace of a hereditarily normal space is hereditarily normal.

Textbook references

 * , Page 205, Exercise 6 (definition introduced in exercise): Introduced using term completely normal space