Paracompact space

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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)