Cover (topology) - Citizendia

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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Set theory, the term Union (denoted as ∪ refers to a set operation used in the convergence of set elements to form a resultant set containing the elements of both sets 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

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

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. In Mathematics, an indexed family of sets is defined in stages beginning with the more general concept of an indexed family of elements, which is really just an alternative Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of 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. In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in 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 Topology and related areas of Mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a Topological space. In Mathematics, a set is called finite if there is a Bijection between the set and some set of the form {1 2. 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 $U_{\alpha \in A}$ 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.

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. In Mathematics, a Lindelöf space is a Topological space in which every Open cover has a countable subcover
metacompact if every open cover has a point finite open refinement. In Mathematics, in the field of General topology, a Topological space is said to be metacompact if every Open cover has a Point finite
paracompact if every open cover admits a locally finite, open refinement. In Mathematics, a paracompact space is a Topological space in which every Open cover admits an open locally finite refinement.

For some more variations see the above articles.

References

Introduction to Topology, Second Edition, Theodore W. In Mathematics, a covering space is a Topological space C which "covers" another space X by a Surjective Local homeomorphism For other uses of "atlas" see Atlas (disambiguation. In Mathematics, particularly topology an atlas describes how In Mathematics, the Lebesgue covering dimension or topological dimension of a Topological space is defined to be the minimum value of n, such 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