Metacompact space

Short description: Topological space with a point-finite open refinement for every cover

In the mathematical field of general topology, a topological space is said to be metacompact if every open cover has a point-finite open refinement. That is, given any open cover of the topological space, there is a refinement that is again an open cover with the property that every point is contained only in finitely many sets of the refining cover.

A space is countably metacompact if every countable open cover has a point-finite open refinement.

Properties

The following can be said about metacompactness in relation to other properties of topological spaces:

• Every paracompact space is metacompact. This implies that every compact space is metacompact, and every metric space is metacompact. The converse does not hold: a counter-example is the Dieudonné plank.
• Every metacompact space is orthocompact.
• Every metacompact normal space is a shrinking space
• The product of a compact space and a metacompact space is metacompact. This follows from the tube lemma.
• An easy example of a non-metacompact space (but a countably metacompact space) is the Moore plane.
• In order for a Tychonoff space X to be compact it is necessary and sufficient that X be metacompact and pseudocompact (see Watson).

Covering dimension

A topological space X is said to be of covering dimension n if every open cover of X has a point-finite open refinement such that no point of X is included in more than n + 1 sets in the refinement and if n is the minimum value for which this is true. If no such minimal n exists, the space is said to be of infinite covering dimension.

See also

• Compact space
• Paracompact space
• Normal space
• Realcompact space
• Pseudocompact space
• Mesocompact space
• Tychonoff space
• Glossary of topology

References

• Watson, W. Stephen (1981). "Pseudocompact metacompact spaces are compact". Proc. Amer. Math. Soc. 81: 151–152. doi:10.1090/s0002-9939-1981-0589159-1. .
• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978]. Counterexamples in Topology (Dover reprint of 1978 ed.). Berlin, New York: Springer-Verlag. ISBN 978-0-486-68735-3.  P.23.