Cover (topology)
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In mathematics, a cover of a set X is a collection of subsets C of X whose union is X. In symbols, if C = {Uα : α ∈ A} is an indexed family of subsets 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 of X whose union contains Y, then C is said to be a cover of Y.
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 Uα 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 = {Vβ : β ∈ B} is a refinement of the cover C = {Uα : α ∈ A} if for any Vβ there exists some Uα such that Vβ ⊆ Uα.
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 = {Uα} is locally finite if for any x ∈ X, there exists some neighborhood N(x) of x such that the set
- <math>\left\{ \alpha \in A : U_{\alpha} \cap N(x) \neq \varnothing \right\}<math>
is finite.
Compactness
The language of covers is often used to define several topological properties related to compactness. A topological space X is said to be
- compact if every open cover has a finite subcover.
- Lindelöf if every open cover has a countable subcover.
- paracompact if every open cover admits a locally finite, open refinement.
For some more variations see the above articles.