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Concept
Cover (topology)
Summary
In mathematics, and more particularly in set theory, a cover (or covering) of a set is a family of subsets of whose union is all of . More formally, if is an indexed family of subsets (indexed by the set ), 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 .
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.
Covers are commonly used in the context of topology. If the set is a topological space, 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 sets in itself or 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 (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 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.
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.
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