Cover (topology)

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In mathematics, a cover of a set X is a collection of sets C such that X is a subset of the union of the sets indexed in C. In symbols, if $C\; =\; lbrace\; U\_alpha:\; alpha\; in\; Arbrace$ is an indexed family of subsets U, of X, then C is a cover if (for definition see: Gamelin and Greene- pg 19, or Kelley- page 49)

$X\; subseteq\; bigcup\_\{alpha\; in\; A\}U\_\{alpha\}$

More generally, if Y is a subset of X, and C is a collection of subsets U_α of X, whose union contains Y, then C is said to be a cover of Y. i.e. C is a cover of Y if

$bigcup\_\{alpha\; in\; A\}U\_\{alpha\}\; supseteq\; Y$

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 U_α 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, $D\; =\; V\_\{beta\; in\; B\}$ is a refinement of $U\_\{alpha\; in\; A\}$ when $forall\; beta\; exists\; alpha\; V\_beta\; subseteq\; U\_alpha$.

Every subcover is also a refinement, but not vice-versa. A subcover is made from the sets that are in the cover, but fewer of them; whereas a refinement is made from any sets that are subsets of the sets in 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 = {U_α} is locally finite if for any x ∈ X, there exists some neighborhood N(x) of x such that the set

$left\{\; alpha\; in\; A\; :\; U\_\{alpha\}\; cap\; N(x)\; neq\; emptyset\; right\}$

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.
• metacompact if every open cover has a point finite open refinement.
• paracompact if every open cover admits a locally finite, open refinement.

For some more variations see the above articles.

See also

• Covering space
• Atlas (topology)
• Lebesgue covering dimension

References

1. Introduction to Topology, Second Edition, Theodore W. Gamelin & Robert Everist Greene. Dover Publications 1999. ISBN: 0-486-40680-6
2. General Topology, John L. Kelley. D. Van Nostrand Company, Inc. Princeton, NJ. 1955.

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