Metacompact space

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

• Compact space
• Paracompact space

Weaker properties

• Orthocompact space
• Countably metacompact space
• MetaLindelof space
• 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 Willard^{More info}, Page 152 (formal definition)