# Metacompact space

In mathematics, in the 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 which 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. MR 507446. P.23.