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

property	quick description	proof of implication	proof of strictness (reverse implication failure)	intermediate notions
Metrizable space	underlying topology of a metric space	metrizable implies perfectly normal	perfectly normal not implies metrizable	|FULL LIST, MORE INFO
CW-space	underlying topology of a CW-complex	CW implies perfectly normal	perfectly normal implies CW	|FULL LIST, MORE INFO

Weaker properties

property	quick description	proof of implication	proof of strictness (reverse implication failure)	intermediate notions
Hereditarily normal space	every subspace is a normal space	perfectly normal implies hereditarily normal	hereditarily normal not implies perfectly normal	|FULL LIST, MORE INFO
Normal space	$T_{1}$ and any two disjoint closed subsets are separated by disjoint open subsets	perfectly normal implies normal	normal not implies perfectly normal	Hereditarily normal space|FULL LIST, MORE INFO
Perfect space	every point is $G_{\delta }$	perfectly normal implies perfect	perfect not implies perfectly normal	|FULL LIST, MORE INFO

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)