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Metacompact space

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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

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

Covering dimension

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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

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References

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  • 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.