In the
mathematical
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
field of
general topology, a
topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
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
In mathematics, a Set (mathematics), set is countable if either it is finite set, finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function fro ...
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
In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
is metacompact. The converse does not hold: a counter-example is the
Dieudonné plank.
* Every metacompact space is
orthocompact.
* Every metacompact
normal space
Normal(s) or The Normal(s) may refer to:
Film and television
* Normal (2003 film), ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* Normal (2007 film), ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keit ...
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
Normal(s) or The Normal(s) may refer to:
Film and television
* Normal (2003 film), ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* Normal (2007 film), ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keit ...
*
Realcompact space
*
Pseudocompact space
*
Mesocompact space
*
Tychonoff space
*
Dowker space
References
*.
* P.23.
{{topology-stub
Properties of topological spaces
Compactness (mathematics)