Cover (topology)

In mathematics, a cover of a set is a collection of sets whose union contains as a subset. Formally, if

is an indexed family of sets , then is a cover of if

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_αcover 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