# Cover (topology)

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In mathematics, a cover of a set X is a collection of sets whose union contains X as a subset. Formally, if

## Contents

C = { U α : α A }

is an indexed family of sets U α , then C is a cover of X if

X α A U α .

## 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 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

Y α A U α

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 xX, there exists some neighborhood N(x) of x such that the set

{ α A : U α 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 = V β B

is a refinement of

U α A when β   α   V β U α .

In other words, there is a refinement map ϕ : B A satisfying V β U ϕ ( β ) for every β B . This map is used, for instance, in the Čech cohomology of X.

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 a 0 < a 1 < . . . < a n being a 0 < b 0 < a 1 < a 2 < . . . < a n < b 1 ), 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. Turn to open cover. Let B be the topological basis of X , we have A = { A B , A U } , where U is any set in an open cover O . A is indeed a refinement. For any A A , we select a U A O (require the selection axiom). Now C = { U A O , A U A } is a subcover of O . Hence the cardinal of a subcover of a open cover can be as small as that of topological basis. And second countability implies Lindelöf spaces.

## 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. If no such minimal n exists, the space is said to be of infinite covering dimension.

## References

Cover (topology) Wikipedia

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