cover (original) (raw)

Definition ([1], pp. 49) Let Y be a subset of a set X. A cover for Y is a collectionof sets 𝒰={Ui}i∈I such that each Uiis a subset of X, and

The collection of sets can be arbitrary, that is, I can be finite, countable, or uncountable. The cover is correspondingly called afinite cover, countable cover, or uncountable cover.

A subcover of 𝒰 is a subset 𝒰′⊂𝒰 such that 𝒰′ is also a cover of X.

A refinement 𝒱 of 𝒰 is a cover of X such that for every V∈𝒱 there is some U∈𝒰 such that V⊂U. When 𝒱 refines 𝒰, it is usually written 𝒱⪯𝒰. ⪯ is a preorder on the set of covers of any topological space X.

If X is a topological space and the members of 𝒰 are open sets, then 𝒰 is said to be an open cover. Open subcovers and open refinements are defined similarly.

Examples

1. 1. If X is a set, then {X} is a cover of X.
2. 1. The power set of a set X is a cover of X.
3. 1. A topology for a set is a cover of that set.

References

• 1 J.L. Kelley, General Topology, D. van Nostrand Company, Inc., 1955.