Paracompact space
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)