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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.
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