# Cover (topology)

# Cover (topology)

`Inmathematics, a`

**cover**of asetis a collection of sets whose union containsas asubset. Formally, if`is anindexed familyof sets, thenis a cover ofif`

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 *U*α of *X* whose union is the whole space *X*. In this case we say that *C* *covers* *X*, or that the sets *U*α *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 *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 which intersects only finitely many sets in the cover. Formally, *C* = {*U*α} 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. 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,

- .

`In other words, there is a`

**refinement map**satisfyingfor every. This map is used, for instance, in theČech cohomologyof 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 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 aninterval(one refinement ofbeing), consideringtopologies(thestandard topologyin euclidean space being a refinement of thetrivial topology). When subdividingsimplicial complexes(the firstbarycentric subdivisionof a simplicial complex is a refinement), the situation is slightly different: everysimplexin 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. Letbe the topological basis of, we have, whereis any set in an open cover.is indeed a refinement. For any, we select a(require the axiom of choice). Nowis a subcover of. Hence the cardinal of a subcover of an 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

- Compactif every open cover has a finite subcover, (or equivalently that every open cover has a finite refinement);Lindelöfif every open cover has acountablesubcover, (or equivalently that every open cover has a countable refinement);Metacompactif every open cover has a point finite open refinement;Paracompactif 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)

Covering space

Partition of a set

Set cover problem

## References

*Differential Forms in Algebraic Topology*. p. 111.

*Topology*(2nd ed.). Prentice Hall. ISBN 0-13-181629-2.