Cover (topology)

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In mathematics, a cover of a set X is a collection of sets C whose union is X. In symbols, if <math>C = \lbrace U_\alpha: \alpha \in A\rbrace</math> is an indexed family of subsets U, of X, then C is a cover if

<math>\bigcup_{\alpha \in A}U_{\alpha} = X</math>

More generally, if Y is a subset of X, and C is a collection of subsets U_α of X, whose union contains Y, then C is said to be a cover of Y. i.e. C is a cover of Y if

<math>\bigcup_{\alpha \in A}U_{\alpha} \supseteq Y </math>

Covers are commonly used in the context of topology. If the set X is a topological space, we say that C is an open cover if each of its members are open sets (i.e. each U_α is contained in T, where T is the topology on X).

If C is a cover of X then a subcover of C is a subset of C which still covers X.

A refinement of a cover C of X is a new cover D of X such that every set in D is contained in some set in C. In symbols, the cover D = {V_β : β ∈ B} is a refinement of the cover C = {U_α : α ∈ A} if for every V_β there exists some U_α such that V_β ⊆ U_α.

Every subcover is also a refinement, but not vice-versa. Note however that a refinement will, in general, have more sets than the original cover.

An open 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. In symbols, C = {U_α} is locally finite if for any x ∈ X, there exists some neighborhood N(x) of x such that the set