Metacompact space: Difference between revisions
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===Stronger properties=== | ===Stronger properties=== | ||
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! property !! quick description !! 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 || || || {{intermediate notions short|metacompact space|compact space}} | |||
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| [[Weaker than::Paracompact space]] || every open cover has a locally finite open refinement || [[paracompact implies metacompact]] || [[metacompact not implies paracompact]] || {{intermediate notions short|metacompact space|paracompact space}} | |||
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| [[Weaker than::Hereditarily metacompact space]] || every subspace is metacompact || || [[metacompactness is not hereditary]] || {{intermediate notions short|metacompact space|hereditarily metacompact space}} | |||
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===Weaker properties=== | ===Weaker properties=== | ||
{| class="wikitable" border="1" | |||
! property !! quick description !! proof of implication !! proof of strictness (reverse implication failure) !! intermediate notions | |||
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| [[Stronger than::Orthocompact space]] || || || || {{intermediate notions short|orthocompact space|metacompact space}} | |||
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| [[Stronger than::Countably metacompact space]] || every countable open cover has a point-finite open refinement || || [[countably metacompact not implies metacompact]] || {{intermediate notions short|countably metacompact space|metacompact space}} | |||
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| [[Stronger than::MetaLindelof space]] || every open cover has a point-countable open refinement || || || | |||
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| [[Stronger than::Nearly metacompact space]] || || || || | |||
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==Metaproperties== | ==Metaproperties== | ||
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A direct product of metacompact spaces need not be metacompact. However, it follows from the [[tube lemma]] that a direct product of a metacompact space with a compact space is metacompact. | A direct product of metacompact spaces need not be metacompact. However, it follows from the [[tube lemma]] that a direct product of a metacompact space with a compact space is metacompact. | ||
==References== | |||
===Textbook references=== | |||
* {{booklink|Willard}}, Page 152 (formal definition) | |||
Latest revision as of 15:45, 25 October 2009
This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces
This is a variation of compactness. View other variations of compactness
Definition
A topological space is said to be metacompact if it satisfies the following property: every open cover has a point-finite open refinement.
Relation with other properties
Stronger properties
| property | quick description | proof of implication | proof of strictness (reverse implication failure) | intermediate notions |
|---|---|---|---|---|
| Compact space | every open cover has a finite subcover | Paracompact space|FULL LIST, MORE INFO | ||
| Paracompact space | every open cover has a locally finite open refinement | paracompact implies metacompact | metacompact not implies paracompact | |FULL LIST, MORE INFO |
| Hereditarily metacompact space | every subspace is metacompact | metacompactness is not hereditary | |FULL LIST, MORE INFO |
Weaker properties
| property | quick description | proof of implication | proof of strictness (reverse implication failure) | intermediate notions |
|---|---|---|---|---|
| Orthocompact space | |FULL LIST, MORE INFO | |||
| Countably metacompact space | every countable open cover has a point-finite open refinement | countably metacompact not implies metacompact | |FULL LIST, MORE INFO | |
| MetaLindelof space | every open cover has a point-countable open refinement | |||
| Nearly metacompact space |
Metaproperties
Products
NO: This property of topological spaces is not a product-closed property of topological spaces: a product of topological spaces, each satisfying the property, when equipped with the product topology, does not necessarily satisfy the property.
View other properties that are not product-closed
A direct product of metacompact spaces need not be metacompact. However, it follows from the tube lemma that a direct product of a metacompact space with a compact space is metacompact.
References
Textbook references
- General topology by Stephen WillardMore info, Page 152 (formal definition)