Metacompact space
From Topospaces
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
Contents
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)
