Cover (topology)

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In mathematics, a cover of a set X is a collection of sets such that X is a subset of the union of sets in the collection. Formally, if

C = \lbrace U_\alpha: \alpha \in A\rbrace

is an indexed family of sets Uα, then C is a cover of X if

X \subseteq \bigcup_{\alpha \in A}U_{\alpha}

Contents

[edit] Cover in topology

Covers are commonly used in the context of topology. If the set X is a topological space, then a cover C of X is a collection of subsets Uα of X whose union is the whole space X. In this case we say that C covers X, or that the sets Uα cover X. Also, if Y is a subset of X, then a cover of Y is a collection of subsets of X whose union contains Y, i.e., C is a cover of Y if

Y \subseteq \bigcup_{\alpha \in A}U_{\alpha}

Let C be a cover of a topological space X. A subcover of C is a subset of C that still covers X.

We say that C is an open cover if each of its members is an open set (i.e. each Uα is contained in T, where T is the topology on X).

A 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. Formally, C = {Uα} 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 \varnothing \right\}

is finite. A cover of X is said to be point finite if every point of X is contained in only finitely many sets in the cover.

[edit] Refinement

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. Formally,

D = V_{\beta \in B}

is a refinement of

U_{\alpha \in A} \qquad \mbox{when} \qquad \forall \beta \ \exists \alpha \ V_\beta \subseteq U_\alpha.

Every subcover is also a refinement, but the opposite is not always true. 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 the cover.

The refinement relation is a preorder on the set of covers of X.

Another useful notion of refinement is star refinement.

[edit] 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. This is equivalent to the requirement that every open cover have a finite refinement.
  • Lindelöf if every open cover has a countable subcover. This is equivalent to the requirement that every open cover has a countable refinement.
  • 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.

[edit] See also

[edit] 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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