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. Cover in topology 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. Refinement 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