In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, in the field of general topology

General topology

In mathematics, general topology or point-set topology is the branch of topology which studies properties of topological spaces and structures defined on them...

, a topological space

Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

 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

Countable set

In mathematics, a countable set is a set with the same cardinality as some subset of the set of natural numbers. A set that is not countable is called uncountable. The term was originated by Georg Cantor...

 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
  Paracompact space
  In mathematics, a paracompact space is a topological space in which every open cover admits a locally finite open refinement. Paracompact spaces are sometimes also required to be Hausdorff. Paracompact spaces were introduced by ....
  
   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
  Normal space
  In topology and related branches of mathematics, a normal space is a topological space X that satisfies Axiom T4: every two disjoint closed sets of X have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space...
  
   is a shrinking space
  Shrinking space
  In mathematics, in the field of topology, a topological space is said to be a shrinking space if every open cover admits a shrinking. A shrinking of an open cover is another open cover indexed by the same indexing set, with the property that the closure of each open set in the shrinking lies inside...
  
  
• The product of a compact space
  Compact space
  In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
  
   and a metacompact space is metacompact. This follows from the tube lemma
  Tube lemma
  In mathematics, particularly topology, the tube lemma is a useful tool in order to prove that the finite product of compact spaces is compact. It is in general, a concept of point-set topology.-Tube lemma:...
  
  .
• An easy example of a non-metacompact space (but a countably metacompact space) is the Moore plane
  Moore plane
  In mathematics, the Moore plane, also sometimes called Niemytzki plane is a topological space. It is a completely regular Hausdorff space which is not normal...
  
  .
• In order that a Tychonoff space
  Tychonoff space
  In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces.These conditions are examples of separation axioms....
  
   X is compact it is necessary and sufficient that X is 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
  Compact space
  In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
  
  
• Paracompact space
  Paracompact space
  In mathematics, a paracompact space is a topological space in which every open cover admits a locally finite open refinement. Paracompact spaces are sometimes also required to be Hausdorff. Paracompact spaces were introduced by ....
  
  
• Normal space
  Normal space
  In topology and related branches of mathematics, a normal space is a topological space X that satisfies Axiom T4: every two disjoint closed sets of X have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space...
  
  
• Realcompact space
  Realcompact space
  In mathematics, in the field of topology, a topological space is said to be realcompact if it is completely regular Hausdorff and every point of its Stone-Cech compactification is real...
  
  
• Pseudocompact space
  Pseudocompact space
  In mathematics, in the field of topology, a topological space is said to be pseudocompact if its image under any continuous function to R is bounded.-Properties related to pseudocompactness:...
  
  
• Mesocompact space
  Mesocompact space
  In mathematics, in the field of general topology, a topological space is said to be mesocompact if every open cover has a compact-finite open refinement...
  
  
• Tychonoff space
  Tychonoff space
  In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces.These conditions are examples of separation axioms....