Paracompact space: Difference between revisions
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[[: | * {{#ask: [[Variation of::Paracompact space]]|limit = 0|searchlabel = Variations of paracompactness}} | ||
===Stronger properties=== | ===Stronger properties=== | ||
{| class="sortable" border="1" | |||
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
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| [[Weaker than::compact space]] || every open cover has a finite subcover || [[compact implies paracompact]] || [[paracompact not implies compact]] || {{intermediate notions short|paracompact space|compact space}} | |||
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| [[Weaker than::hereditarily paracompact space]] || every subspace is paracompact || || || {{intermediate notions short|paracompact space|hereditarily paracompact space}} | |||
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| [[Weaker than::strongly paracompact space]] || every open cover has a star-finite open refinement || || || {{intermediate notions short|paracompact space|strongly paracompact space}} | |||
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| [[Weaker than::paracompact Hausdorff space]] || paracompact and [[Hausdorff space|Hausdorff]] || || || {{intermediate notions short|paracompact space|paracompact Hausdorff space}} | |||
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| [[Weaker than::regular Lindelof space]] || [[regular space|regular]] and [[Lindelof space|Lindelof]] || || || {{intermediate notions short|paracompact space|regular Lindelof space}} | |||
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| [[Weaker than::metrizable space]] || underlying topology of a [[metric space]] || || || {{intermediate notions short|paracompact space|metrizable space}} | |||
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| [[Weaker than::manifold]] || || || || {{intermediate notions short|paracompact space|manifold}} | |||
|} | |||
===Weaker properties=== | ===Weaker properties=== | ||
{| class="sortable" border="1" | |||
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
|- | |||
| [[Stronger than::metacompact space]] || every open cover has a point-finite open refinement || [[paracompact implies metacompact]] || [[metacompact not implies paracompact]] || {{intermediate notions short|metacompact space|paracompact space}} | |||
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| [[Stronger than::orthocompact space]] || || (via metacompact)|| (via metacompact) || {{intermediate notions short|orthocompact space|paracompact space}} | |||
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| [[Stronger than::locally paracompact space]] || || || || | |||
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| [[Stronger than::countably paracompact space]] || || || || | |||
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==Metaproperties== | ==Metaproperties== | ||
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{{further|[[compact times paracompact implies paracompact]]}} | {{further|[[compact times paracompact implies paracompact]]}} | ||
==References== | |||
===Textbook references=== | |||
* {{booklink|Munkres}}, Page 253 (formal definition) | |||
* {{booklink|SingerThorpe}}, Page 148 (formal definition) | |||
Latest revision as of 02:18, 21 October 2010
This article defines a property of topological space that is pivotal (viz important) among currently studied properties of topological spaces
This is a variation of compactness. View other variations of compactness
Definition
A topological space is said to be paracompact if it satisfies the following condition: every open cover has a locally finite open refinement.
Relation with other properties
This property is a pivotal (important) member of its property space. Its variations, opposites, and other properties related to it and defined using it are often studied
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| compact space | every open cover has a finite subcover | compact implies paracompact | paracompact not implies compact | |FULL LIST, MORE INFO |
| hereditarily paracompact space | every subspace is paracompact | |FULL LIST, MORE INFO | ||
| strongly paracompact space | every open cover has a star-finite open refinement | |FULL LIST, MORE INFO | ||
| paracompact Hausdorff space | paracompact and Hausdorff | |FULL LIST, MORE INFO | ||
| regular Lindelof space | regular and Lindelof | |FULL LIST, MORE INFO | ||
| metrizable space | underlying topology of a metric space | |FULL LIST, MORE INFO | ||
| manifold | |FULL LIST, MORE INFO |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| metacompact space | every open cover has a point-finite open refinement | paracompact implies metacompact | metacompact not implies paracompact | |FULL LIST, MORE INFO |
| orthocompact space | (via metacompact) | (via metacompact) | Metacompact space|FULL LIST, MORE INFO | |
| locally paracompact space | ||||
| countably paracompact space |
Metaproperties
Hereditariness
This property of topological spaces is not hereditary on all subsets
A paracompact space can have non-paracompact subspaces.
Weak hereditariness
This property of topological spaces is weakly hereditary or closed subspace-closed; in other words, any closed subset (equipped with the subspace topology) of a space with the property, also has the property.
View all weakly hereditary properties of topological spaces | View all subspace-hereditary properties of topological spaces
Any closed subspace of a paracompact space is paracompact.
Effect of property modifiers
The product-transiter
Applying the product-transiter to this property gives: product-transitively paracompact space
Although a product of paracompact spaces need not be paracompact, there is a subclass of paracompact spaces with which the product of any paracompact space is paracompact. Such spaces are termed product-transitively paracompact; all compact spaces are product-transitively paracompact.
Further information: compact times paracompact implies paracompact
References
Textbook references
- Topology (2nd edition) by James R. MunkresMore info, Page 253 (formal definition)
- Lecture Notes on Elementary Topology and Geometry (Undergraduate Texts in Mathematics) by I. M. Singer and J. A. ThorpeMore info, Page 148 (formal definition)