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 is an indexed family of sets U_α, then C is a cover of X if For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... Superset redirects here. ... In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ... In mathematics, an index set is another name for a function domain. ...

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 For other uses, see Topology (disambiguation). ...

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). In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U...

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 This is a glossary of some terms used in the branch of mathematics known as topology. ... In mathematics, a set is called finite if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ...

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, is a refinement of when .

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.
• 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. In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... In mathematics, a Lindelöf space is a topological space in which every open cover has a countable subcover. ... In mathematics the term countable set is used to describe the size of a set, e. ... In mathematics, a paracompact space is a topological space in which every open cover admits an open locally finite refinement. ...

See also

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

In mathematics, specifically topology, a covering map is a continuous surjective map p : C → X, with C and X being topological spaces, which has the following property: to every x in X there exists an open neighborhood U such that p -1(U) is a union of mutually disjoint open... In topology, a branch of mathematics, an atlas describes how a complicated space called a manifold is glued together from simpler pieces. ... In mathematics, the Lebesgue covering dimension or topological dimension of a topological space is defined to be the minimum value of n, such that any open cover has a refinement in which no point is included in more than n+1 elements. ...

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.