In the
mathematical
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
field of
general topology
In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometri ...
, a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
is said to be metacompact if every
open cover
In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alp ...
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 is countable if either it is 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 from it into the natural numbers ...
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
In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is normal ...
is metacompact. This implies that every
compact space
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...
is metacompact, and every
metric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
is metacompact. The converse does not hold: a counter-example is the
Dieudonné plank.
* Every metacompact space is
orthocompact
In mathematics, in the field of general topology, a topological space is said to be orthocompact if every open cover has an interior-preserving open refinement.
That is, given an open cover of the topological space, there is a refinement that is ...
.
* Every metacompact
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. T ...
is a
shrinking space
* The product of a
compact space
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...
and a metacompact space is metacompact. This follows from the
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.
Statement
The lemma uses the following terminology:
* If X and Y are topological spaces and X \times ...
.
* An easy example of a non-metacompact space (but a countably metacompact space) is the
Moore plane In mathematics, the Moore plane, also sometimes called Niemytzki plane (or Nemytskii plane, Nemytskii's tangent disk topology), is a topological space. It is a completely regular Hausdorff space (also called Tychonoff space) that is not normal. I ...
.
* In order for a
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. A Tychonoff space refers to any completely regular space that is ...
''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
In mathematics, the Lebesgue covering dimension or topological dimension of a topological space is one of several different ways of defining the dimension of the space in a
topologically invariant way.
Informal discussion
For ordinary Euclide ...
''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
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...
*
Paracompact space
In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is normal ...
*
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. T ...
*
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–Čech compactification is real (meaning that the quotient field at that point of the ri ...
*
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. Many authors include the requirement that the space be completely regular in the definition of ps ...
*
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. That is, given any open cover, we can find an open refinement with the property that ever ...
*
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. A Tychonoff space refers to any completely regular space that is ...
*
Glossary of topology
This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology. The following definitions are also fun ...
References
*.
* P.23.
{{topology-stub
Properties of topological spaces
Compactness (mathematics)