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cover∗ mps† 2013-03-21 13:04:11 Definition ([?], pp. 49) Let Y be a subset of a set X. A cover for Y is a collection of sets U = {Ui }i∈I such that each Ui is a subset of X, and [ Y ⊂ Ui . i∈I The collection of sets can be arbitrary, that is, I can be finite, countable, or uncountable. The cover is correspondingly called a finite cover, countable cover, or uncountable cover. A subcover of U is a subset U 0 ⊂ U such that U 0 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. If X is a set, then {X} is a cover of X. 2. The power set of a set X is a cover of X. 3. A topology for a set is a cover of that set. References [1] J.L. Kelley, General Topology, D. van Nostrand Company, Inc., 1955. ∗ hCoveri created: h2013-03-21i by: hmpsi version: h31224i Privacy setting: h1i hDefinitioni h54A99i † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. 1