Perfectly normal space

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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. MunkresMore info, Page 213, Exercise 6 (definition introduced in exercise)