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


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

A subcover of 𝒰 is a subset 𝒰â€Č⊂𝒰 such that 𝒰â€Č is also a cover of X.

A refinement đ’± of 𝒰 is a cover of X such that for every Vâˆˆđ’± there is some U∈𝒰 such that V⊂U. When đ’± refines 𝒰, it is usually written đ’±âȘŻđ’°. âȘŻ is a preorder on the set of covers of any topological spaceMathworldPlanetmath 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.


  1. 1.

    If X is a set, then {X} is a cover of X.

  2. 2.

    The power setMathworldPlanetmath of a set X is a cover of X.

  3. 3.

    A topology for a set is a cover of that set.


  • 1 J.L. Kelley, General Topology, D. van Nostrand Company, Inc., 1955.
Title cover
Canonical name Cover
Date of creation 2013-03-22 12:06:31
Last modified on 2013-03-22 12:06:31
Owner mps (409)
Last modified by mps (409)
Numerical id 19
Author mps (409)
Entry type Definition
Classification msc 54A99
Related topic CompactPlanetmathPlanetmath
Related topic VarepsilonNet
Related topic Site
Related topic CoveringSpace
Related topic CompactMetricSpacesAreSecondCountable
Defines open cover
Defines subcover
Defines refinement
Defines finite cover
Defines countable cover
Defines uncountable cover
Defines open subcover
Defines open refinement
Defines cover refinement