Perfectly normal space

In the T family (properties of topological spaces related to separation axioms), this is called: T6

This is a variation of normality. View other variations of normality

Definition

A topological space is termed perfectly normal if it is normal and every closed subset is a G-delta subset ($G_{\delta }$).

Formalisms

Subspace property implication formalism

This property of topological spaces can be encoded by the fact that one subspace property implies another

Modulo the assumption of the space being T1, the property of being perfectly normal can be encoded as:

Closed $\implies$ $G_{\delta }$

Relation with other properties

Stronger properties

• Metrizable space
• CW-space

Weaker properties

• Hereditarily normal space
• Normal space
• Perfect 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

Any subspace of a perfectly normal space is perfectly normal.

References

Textbook references

• Topology (2nd edition) by James R. Munkres^{More info}, Page 213, Exercise 6 (definition introduced in exercise)