cover

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 .