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 Dowker space
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 0507446. P.23.