In
mathematics
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 ...
, and more particularly in
set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...
, a cover (or covering) of a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
is a collection of
subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
s of
whose union is all of
. More formally, if
is an
indexed family of subsets
, then
is a cover of
if
. Thus the collection
is a cover of
if each element of
belongs to at least one of the subsets
.
Cover in topology
Covers are commonly used in the context of
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
. If the set
is 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 ...
, then a ''cover''
of
is a collection of subsets
of
whose union is the whole space
. In this case we say that
''covers''
, or that the sets
''cover''
.
Also, if
is a (topological) subspace of
, then a ''cover'' of
is a collection of subsets
of
whose union contains
, i.e.,
is a cover of
if
:
That is, we may cover
with either open sets in
itself, or cover
by open sets in the parent space
.
Let ''C'' be a cover of a topological space ''X''. A subcover of ''C'' is a subset of ''C'' that still covers ''X''.
We say that ''C'' is an if each of its members is an
open set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
(i.e. each ''U''
''α'' is contained in ''T'', where ''T'' is the topology on ''X'').
A cover of ''X'' is said to be
locally finite if every point of ''X'' has a
neighborhood that intersects only
finitely many sets in the cover. Formally, ''C'' = is locally finite if for any
there exists some neighborhood ''N''(''x'') of ''x'' such that the set
:
is finite. A cover of ''X'' is said to be point finite if every point of ''X'' is contained in only finitely many sets in the cover. A cover is point finite if it is locally finite, though the converse is not necessarily true.
Refinement
A refinement of a cover
of a topological space
is a new cover
of
such that every set in
is contained in some set in
. Formally,
:
is a refinement of
if for all
there exists
such that
In other words, there is a refinement map
satisfying
for every
This map is used, for instance, in the
Čech cohomology of
.
Every subcover is also a refinement, but the opposite is not always true. A subcover is made from the sets that are in the cover, but omitting some of them; whereas a refinement is made from any sets that are subsets of the sets in the cover.
The refinement relation is a
preorder on the set of covers of
.
Generally speaking, a refinement of a given structure is another that in some sense contains it. Examples are to be found when partitioning an
interval (one refinement of
being
), considering
topologies (the
standard topology in euclidean space being a refinement of the
trivial topology). When subdividing
simplicial complexes (the first
barycentric subdivision
In mathematics, the barycentric subdivision is a standard way to subdivide a given simplex into smaller ones. Its extension on simplicial complexes is a canonical method to refine them. Therefore, the barycentric subdivision is an important tool i ...
of a simplicial complex is a refinement), the situation is slightly different: every
simplex
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
in the finer complex is a face of some simplex in the coarser one, and both have equal underlying polyhedra.
Yet another notion of refinement is that of
star refinement.
Subcover
A simple way to get a subcover is to omit the sets contained in another set in the cover. Consider specifically open covers.
Let
be a topological basis of
and
be an open cover of
First take
Then
is a refinement of
. Next, for each
we select a
containing
(requiring the axiom of choice). Then
is a subcover of
Hence the cardinality of a subcover of an open cover can be as small as that of any topological basis. Hence in particular second countability implies a space is
Lindelöf.
Compactness
The language of covers is often used to define several topological properties related to ''compactness''. A topological space ''X'' is said to be
;
Compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in Britis ...
: if every open cover has a finite subcover, (or equivalently that every open cover has a finite refinement);
;
Lindelöf: if every open cover has a
countable subcover, (or equivalently that every open cover has a countable refinement);
;
Metacompact: if every open cover has a point-finite open refinement;
;
Paracompact: if every open cover admits a locally finite open refinement.
For some more variations see the above articles.
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
*
*
*
*
*
*
*
Notes
References
#''Introduction to Topology'', Second Edition, Theodore W. Gamelin & Robert Everist Greene. Dover Publications 1999.
#''General Topology'',
John L. Kelley. D. Van Nostrand Company, Inc. Princeton, NJ. 1955.
External links
* {{springer, title=Covering (of a set), id=p/c026950
Topology
General topology