Perfectly normal space: Difference between revisions
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A [[topological space]] is termed '''perfectly normal''' if it is [[normal space|normal]] and every closed subset is a [[G-delta subset]] (<math>G_\delta</math>). | A [[topological space]] is termed '''perfectly normal''' if it is [[normal space|normal]] and every closed subset is a [[G-delta subset]] (<math>G_\delta</math>). | ||
==Formalisms== | |||
{{subspace property implication}} | |||
Modulo the assumption of the space being [[T1 space|T1]], the property of being perfectly normal can be encoded as: | |||
Closed <math>\implies</math> <math>G_\delta</math> | |||
==Relation with other properties== | ==Relation with other properties== | ||
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===Stronger properties=== | ===Stronger properties=== | ||
{| class="wikitable" border="1" | |||
! property !! quick description !! proof of implication !! proof of strictness (reverse implication failure) !! intermediate notions | |||
|- | |||
| [[Weaker than::Metrizable space]] || underlying topology of a [[metric space]] || [[metrizable implies perfectly normal]] || [[perfectly normal not implies metrizable]] || {{intermediate notions short|perfectly normal space|metrizable space}} | |||
|- | |||
| [[Weaker than::CW-space]] || underlying topology of a [[CW-complex]] || [[CW implies perfectly normal]] || [[perfectly normal implies CW]] || {{intermediate notions short|perfectly normal space|CW-space}} | |||
|} | |||
===Weaker properties=== | ===Weaker properties=== | ||
{| class="wikitable" border="1" | |||
! property !! quick description !! proof of implication !! proof of strictness (reverse implication failure) !! intermediate notions | |||
|- | |||
| [[Stronger than::Hereditarily normal space]] || every subspace is a [[normal space]] || [[perfectly normal implies hereditarily normal]] || [[hereditarily normal not implies perfectly normal]] || {{intermediate notions short|hereditarily normal space|perfectly normal space}} | |||
|- | |||
| [[Stronger than::Normal space]] || <math>T_1</math> and any two disjoint closed subsets are separated by disjoint open subsets || [[perfectly normal implies normal]] || [[normal not implies perfectly normal]] || {{intermediate notions short|normal space|perfectly normal space}} | |||
|- | |||
| [[Stronger than::Perfect space]] || every point is <math>G_\delta</math> || [[perfectly normal implies perfect]] || [[perfect not implies perfectly normal]] || {{intermediate notions short|perfect space|perfectly normal space}} | |||
|} | |||
==Metaproperties== | ==Metaproperties== | ||
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Any subspace of a perfectly normal space is perfectly normal. | Any subspace of a perfectly normal space is perfectly normal. | ||
==References== | |||
===Textbook references=== | |||
* {{booklink|Munkres}}, Page 213, Exercise 6 (definition introduced in exercise) | |||
Latest revision as of 16:10, 26 October 2009
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 ().
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
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 | 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 | 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)