In mathematics, and more particularly in set theory, a cover (or covering) of a set X {\displaystyle X} is a family of subsets of X {\displaystyle X} whose union is all of X {\displaystyle X} . More formally, if C = { U α : α ∈ A } {\displaystyle C=\lbrace U_{\alpha }:\alpha \in A\rbrace } is an indexed family of subsets U α ⊂ X {\displaystyle U_{\alpha }\subset X} (indexed by the set A {\displaystyle A} ), then C {\displaystyle C} is a cover of X {\displaystyle X} if
⋃ α ∈ A U α = X . {\displaystyle \bigcup _{\alpha \in A}U_{\alpha }=X.}
Thus the collection { U α : α ∈ A } {\displaystyle \lbrace U_{\alpha }:\alpha \in A\rbrace } is a cover of X {\displaystyle X} if each element of X {\displaystyle X} belongs to at least one of the subsets U α {\displaystyle U_{\alpha }} .
Definition Covers are commonly used in the context of topology. If the set X {\displaystyle X} is a topological space, then a cover C {\displaystyle C} of X {\displaystyle X} is a collection of subsets { U α } α ∈ A {\displaystyle \{U_{\alpha }\}_{\alpha \in A}} of X {\displaystyle X} whose union is the whole space X = ⋃ α ∈ A U α {\displaystyle X=\bigcup _{\alpha \in A}U_{\alpha }} . In this case C {\displaystyle C} is said to cover X {\displaystyle X} , or that the sets U α {\displaystyle U_{\alpha }} cover X {\displaystyle X} . If Y {\displaystyle Y} is a (topological) subspace of X {\displaystyle X} , then a cover of Y {\displaystyle Y} is a collection of subsets C = { U α } α ∈ A {\displaystyle C=\{U_{\alpha }\}_{\alpha \in A}} of X {\displaystyle X} whose union contains Y {\displaystyle Y} . That is, C {\displaystyle C} is a cover of Y {\displaystyle Y} if
Y ⊆ ⋃ α ∈ A U α . {\displaystyle Y\subseteq \bigcup _{\alpha \in A}U_{\alpha }.}
Here, Y {\displaystyle Y} may be covered with either sets in Y {\displaystyle Y} itself or sets in the parent space X {\displaystyle X} . A cover of X {\displaystyle X} is said to be locally finite if every point of X {\displaystyle X} has a neighborhood that intersects only finitely many sets in the cover. Formally, C = { U α } {\displaystyle C=\{U_{\alpha }\}} is locally finite if, for any x ∈ X {\displaystyle x\in X} , there exists some neighborhood N ( x ) {\displaystyle N(x)} of x {\displaystyle x} such that the set
{ α ∈ A : U α ∩ N ( x ) ≠ ∅ } {\displaystyle \left\{\alpha \in A:U_{\alpha }\cap N(x)\neq \varnothing \right\}}
is finite. A cover of X {\displaystyle X} is said to be point finite if every point of X {\displaystyle X} is contained in only finitely many sets in the cover. A cover is point finite if locally finite, though the converse is not necessarily true.
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