Cover (topology)
Encyclopedia
In mathematics
, a cover of a set X is a collection of sets whose union contains X as a subset
. Formally, if
is an indexed family
of sets Uα, then C is a cover of X if
. If the set X is a topological space
, then a cover C of X is a collection of subsets Uα of X whose union is the whole space X. In this case we say that C covers X, or that the sets Uα cover X. Also, if Y is a subset of X, then a cover of Y is a collection of subsets of X whose union contains Y, i.e., C is a cover of Y if
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 open cover if each of its members is an open set
(i.e. each Uα 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 which intersects only finitely many sets in the cover. Formally, C = {Uα} is locally finite if for any x ∈ X, 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.
is a refinement 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 X.
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 complex
es (the first barycentric subdivision
of a simplicial complex is a refinement), the situation is slightly different: every simplex
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
.
For some more variations see the above articles.
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...
, a cover of a set X is a collection of sets whose union contains X as a subset
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
. Formally, if
is an indexed family
Indexed family
In mathematics, an indexed family is a collection of values that are associated with indexes. For example, a family of real numbers, indexed by the integers is a collection of real numbers, where each integer is associated with one of the real numbers....
of sets Uα, then C is a cover of X if
Cover in topology
Covers are commonly used in the context of topologyTopology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
. If the set X is 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...
, then a cover C of X is a collection of subsets Uα of X whose union is the whole space X. In this case we say that C covers X, or that the sets Uα cover X. Also, if Y is a subset of X, then a cover of Y is a collection of subsets of X whose union contains Y, i.e., C is a cover of Y if
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 open cover if each of its members is an open set
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
(i.e. each Uα is contained in T, where T is the topology on X).
A cover of X is said to be locally finite
Locally finite collection
In the mathematical field of topology, local finiteness is a property of collections of subsets of a topological space. It is fundamental in the study of paracompactness and topological dimension....
if every point of X has a neighborhood which intersects only finitely many sets in the cover. Formally, C = {Uα} is locally finite if for any x ∈ X, 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.
Refinement
A refinement of a cover C of a topological space X is a new cover D of X such that every set in D is contained in some set in C. Formally,is a refinement 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
Preorder
In mathematics, especially in order theory, preorders are binary relations that are reflexive and transitive.For example, all partial orders and equivalence relations are preorders...
on the set of covers of X.
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
Interval (mathematics)
In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...
(one refinement of
being ), considering topologiesTopology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
(the standard topology in euclidean space being a refinement of the trivial topology
Trivial topology
In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such a space is sometimes called an indiscrete space, and its topology sometimes called an indiscrete topology...
). When subdividing simplicial complex
Simplicial complex
In mathematics, a simplicial complex is a topological space of a certain kind, constructed by "gluing together" points, line segments, triangles, and their n-dimensional counterparts...
es (the first barycentric subdivision
Barycentric subdivision
In geometry, the barycentric subdivision is a standard way of dividing an arbitrary convex polygon into triangles, a convex polyhedron into tetrahedra, or, in general, a convex polytope into simplices with the same dimension, by connecting the barycenters of their faces in a specific way.The name...
of a simplicial complex is a refinement), the situation is slightly different: every simplex
Simplex
In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron,...
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
Star refinement
In mathematics, specifically in the study of topology and open covers of a topological space X, a star refinement is a particular kind of refinement of an open cover of X....
.
Compactness
The language of covers is often used to define several topological properties related to compactness. A topological space X is said to be- CompactCompact spaceIn 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...
, if every open cover has a finite subcover, (or equivalently that every open cover has a finite refinement); - LindelöfLindelöf spaceIn mathematics, a Lindelöf space is a topological space in which every open cover has a countable subcover. The Lindelöf property is a weakening of the more commonly used notion of compactness, which requires the existence of a finite subcover....
, if every open cover has a countable subcover, (or equivalently that every open cover has a countable refinement); - MetacompactMetacompact spaceIn mathematics, in the field of general topology, a topological space is said to be metacompact if every open cover has a point finite open refinement...
, if every open cover has a point finite open refinement; - ParacompactParacompact spaceIn 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 ....
, if every open cover admits a locally finite open refinement.
For some more variations see the above articles.