Metacompact space

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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.

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

• Every paracompact 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

[edit] See also

• Mesocompact

[edit] References

• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of 1978 ed.), Berlin, New York: Springer-Verlag, MR507446, ISBN 978-0-486-68735-3  P.23.