Open cover

Open cover
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Open_cover".

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. In symbols, if

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

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

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

content

1 Cover in topology
2 Refinement
3 Compactness
4 See also
5 References

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 U_α of X whose union is the whole space X. In this case we say that C covers X, or that the sets U_α 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

$\bigcup_{\alpha \in A}U_{\alpha} \supseteq Y$

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 U_α 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. 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

$\left\{ \alpha \in A : U_{\alpha} \cap N(x) \neq \varnothing \right\}$

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.

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. In symbols,

$D = V_{\beta \in B}$

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 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.

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

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.

References

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

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Contents

Cover in topology

Refinement

Compactness

See also

References