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

Category:Variations of paracompactness

Stronger properties

• Compact space
• Hereditarily paracompact space
• Strongly paracompact space
• Paracompact Hausdorff space
• Regular Lindelof space
• Metrizable space

Weaker properties

• Metacompact space
• Orthocompact space
• Locally 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. Munkres^{More info}, Page 253 (formal definition)
• Lecture Notes on Elementary Topology and Geometry (Undergraduate Texts in Mathematics) by I. M. Singer and J. A. Thorpe^{More info}, Page 148 (formal definition)