Cover (topology)

In mathematics, a cover of a set X {displaystyle X} is a collection of sets whose union contains X {displaystyle X} as a subset. Formally, if In mathematics, a cover of a set X {displaystyle X} is a collection of sets whose union contains X {displaystyle X} as a subset. Formally, if is an indexed family of sets U α {displaystyle U_{alpha }} , then C {displaystyle C} is a cover of X {displaystyle X} if 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 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 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. 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 ϕ : B → A {displaystyle phi :B ightarrow A} satisfying V β ⊆ U ϕ ( β ) {displaystyle V_{eta }subseteq U_{phi (eta )}} for every β ∈ B {displaystyle eta in B} . This map is used, for instance, in the Čech cohomology of X.