Cover (topology)

In mathematics, particularly topology, 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 C}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 that 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}\}\;{\text{is a refinement of}}\;C=\{U_{\alpha \in A}\}\;{\text{when}}\;\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 $a_{0}<b_{0}<a_{1}<a_{2}<...<a_{n-1}<b_{1}<a_{n}$ ), 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. Consider specifically open covers. Let ${\mathcal {B}}$ be a topological basis of $X$ and ${\mathcal {O}}$ be an open cover of $X$ . First take ${\mathcal {A}}=\{A\in {\mathcal {B}}\mid \exists U\in {\mathcal {O}}:A\subseteq U\}$ . Then ${\mathcal {A}}$ is a refinement of ${\mathcal {O}}$ . Next, for each $A\in {\mathcal {A}}$ , we select a $U_{A}\in {\mathcal {O}}$ containing $A$ (requiring the axiom of choice). Then ${\mathcal {C}}=\{U_{A}\in {\mathcal {O}}\mid A\in {\mathcal {A}}\}$ is a subcover of ${\mathcal {O}}$ . Hence the cardinality of a subcover of an open cover can be as small as that of any topological basis. Hence in particular second countability implies a space is Lindelöf.

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

"Covering (of a set)", Encyclopedia of Mathematics, EMS Press, 2001 [1994]

[1] Bott, Tu (1982). Differential Forms in Algebraic Topology. p. 111.

[2] Munkres, James (1999). Topology (2nd ed.). Prentice Hall. ISBN 0-13-181629-2.

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