Cover (topology)

Short description: Subsets whose union equals the whole set

In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a family of subsets of X whose union is all of X. More formally, if C={Uα:α∈A} is an indexed family of subsets Uα⊂X (indexed by the set A), then C is a cover of X if ⋃α∈AUα=X. Thus the collection {Uα:α∈A} is a cover of X if each element of X belongs to at least one of the subsets Uα.

A subcover of a cover of a set is a subset of the cover that also covers the set. A cover is called an open cover if each of its elements is an open set.

## Contents

* 1 Cover in topology
* 2 Refinement
* 3 Subcover
* 4 Compactness
* 5 Covering dimension
* 6 See also
* 7 Notes
* 8 References
* 9 External links

## Cover in topology

Covers are commonly used in the context of topology. If the set X is a topological space, then a cover C of X is a collection of subsets {Uα}α∈A 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 (topological) subspace of X, then a cover of Y is a collection of subsets C={Uα}α∈A of X whose union contains Y, i.e., C is a cover of Y if

Y⊆⋃α∈AUα.

That is, we may cover Y with either sets in Y itself or sets in the parent space X.

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 that 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

{α∈A:Uα∩N(x)≠∅}

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 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,

D={Vβ}β∈B is a refinement of C={Uα}α∈A if for all β∈B there exists α∈A such that Vβ⊆Uα.

In other words, there is a refinement map ϕ:B->A satisfying Vβ⊆Uϕ(β) for every β∈B. This map is used, for instance, in the Čech cohomology of X.[1]

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 on the set of covers of X is transitive, irreflexive, and asymmetric.

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 a0<a1<⋯<an being a0<b0<a1<a2<⋯<an−1<b1<an), considering topologies (the standard topology in Euclidean space being a refinement of the trivial topology). When subdividing simplicial complexes (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.

## 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 ℬ be a topological basis of X and 𝒪 be an open cover of X. First take 𝒜={A∈ℬ: there exists U∈𝒪 such that A⊆U}. Then 𝒜 is a refinement of 𝒪. Next, for each A∈𝒜, we select a UA∈𝒪 containing A (requiring the axiom of choice). Then 𝒞={UA∈𝒪:A∈𝒜} is a subcover of 𝒪. 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
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 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.[2] If no such minimal n exists, the space is said to be of infinite covering dimension.

## See also

* Atlas (topology) - Set of charts that describes a manifold
* Bornology - Mathematical generalization of boundedness
* Covering space - Type of continuous map in topology
* Grothendieck topology - Mathematical structure
* Partition of a set - Mathematical ways to group elements of a set
* Set cover problem - Classical problem in combinatorics
* Star refinement

## Notes

1. ↑ Bott, Tu (1982). Differential Forms in Algebraic Topology. p. 111.
2. ↑ Munkres, James (1999). Topology (2nd ed.). Prentice Hall. ISBN 0-13-181629-2.

## References

1. Introduction to Topology, Second Edition, Theodore W. Gamelin & Robert Everist Greene. Dover Publications 1999. ISBN 0-486-40680-6
2. General Topology, John L. Kelley. D. Van Nostrand Company, Inc. Princeton, NJ. 1955.

## External links

* Hazewinkel, Michiel, ed. (2001), "Covering (of a set)", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=p/c026950

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