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
References
- {{Cite journal|first=W. Stephen|last=Watson|title=Pseudocompact metacompact spaces are compact|journal=Proc. Amer. Math. Soc.|volume=81|pages=151–152|year=1981|doi=10.1090/s0002-9939-1981-0589159-1|doi-access=free}}.
- {{Cite book | last1=Steen | first1=Lynn Arthur | author1-link=Lynn Arthur Steen | last2=Seebach | first2=J. Arthur Jr. | author2-link=J. Arthur Seebach, Jr. | title=Counterexamples in Topology | title-link=Counterexamples in Topology | origyear=1978 | publisher=Springer-Verlag | location=Berlin, New York | edition=Dover reprint of 1978 | isbn=978-0-486-68735-3 |mr=507446 | year=1995 }} P.23.
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