Cover (topology)

In mathematics, a cover of a set $X$ is a collection of sets whose union contains $X$ as a subset. Formally, if

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

is an indexed family of sets $U_{\alpha }$ , then $C$ is a cover of $X$ if

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

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

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 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. Formally, 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. A cover is point finite if it is locally finite, though the converse is not necessarily true.

A refinement of a cover C of a topological space 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 }

In other words, there is a refinement map $\phi :B\rightarrow A$ satisfying $V_{{\beta }}\subseteq U_{{\phi (\beta )}}$ for every $\beta \in B$ . This map is used, for instance, in the Čech cohomology of X.^[1]

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 omitting some 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.

Generally speaking, a refinement of a given structure is another that in some sense contains it. Examples are to be found when partitioning an interval (one refinement of $a_{0}<a_{1}<...<a_{n}$ being $a_{0}<b_{0}<a_{1}<a_{2}<...<a_{n}<b_{1}$ ), considering topologies (the standard topology in euclidean space being a refinement of the trivial topology). When subdividing simplicial complexes (the first barycentric subdivision of a simplicial complex is a refinement), the situation is slightly different: every simplex in the finer complex is a face of some simplex in the coarser one, and both have equal underlying polyhedra.

Yet another notion of refinement is that of star refinement.

Subcover

A simple way to get a subcover is to omit the sets contained in another set in the cover. Turn to open cover. Let ${\mathcal {B}}$ be the topological basis of $X$ , we have ${\mathcal {A}}=\{A\in {\mathcal {B}},A\subset U\}$ , where $U$ is any set in an open cover ${\mathcal {O}}$ . ${\mathcal {A}}$ is indeed a refinement. For any $A\in {\mathcal {A}}$ , we select a $U_{A}\in {\mathcal {O}}$ (require the selection axiom). Now ${\mathcal {C}}=\{U_{A}\in {\mathcal {O}},A\subset U_{A}\}$ is a subcover of ${\mathcal {O}}$ . Hence the cardinal of a subcover of an open cover can be as small as that of topological basis. And second countability implies Lindelöf spaces.

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, (or equivalently that every open cover has a finite refinement);
Lindelöf: if every open cover has a countable subcover, (or equivalently 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.

Covering dimension

A topological space X is said to be of covering dimension n if every open cover of X has a point finite open refinement such that no point of X is included in more than n+1 sets in the refinement and if n is the minimum value for which this is true.^[2] If no such minimal n exists, the space is said to be of infinite covering dimension.

Notes

↑ Bott, Tu (1982). Differential Forms in Algebraic Topology. p. 111.
↑ Munkres, James (1999). Topology (2nd ed.). Prentice Hall. ISBN 0-13-181629-2.

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.

External links

Hazewinkel, Michiel, ed. (2001) [1994], "Covering (of a set)", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4

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