Metacompact space

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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)