Cover (topology)

In mathematics, and more particularly in set theory, a cover (or covering) of a set $X$ is a family of subsets of $X$ whose union is all of $X$ . More formally, if $C=\lbrace U_{\alpha }:\alpha \in A\rbrace$ is an indexed family of subsets $U_{\alpha }\subset X$ (indexed by the set $A$ ), then $C$ is a cover of $X$ if $\bigcup _{\alpha \in A}U_{\alpha }\supseteq X$ . Thus the collection $\lbrace U_{\alpha }:\alpha \in A\rbrace$ is a cover of $X$ if each element of $X$ belongs to at least one of the subsets $U_{\alpha }$ .

A subcover of a cover of a set is a subset of the cover that also covers the set. A cover is called an open cover if each of its elements is an open set.

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 $\{U_{\alpha }\}_{\alpha \in A}$ of $X$ whose union is the whole space $X$ . In this case we say that $C$ covers $X$ , or that the sets $U_{\alpha }$ cover $X$ .

Also, if $Y$ is a (topological) subspace of $X$ , then a cover of $Y$ is a collection of subsets $C=\{U_{\alpha }\}_{\alpha \in A}$ of $X$ whose union contains $Y$ , i.e., $C$ is a cover of $Y$ if

Y\subseteq \bigcup _{\alpha \in A}U_{\alpha }.

That is, we may cover $Y$ with either sets in $Y$ itself or sets in the parent space $X$ .

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 $x\in X,$ 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 }\}_{\beta \in B}

is a refinement of

C=\{U_{\alpha }\}_{\alpha \in A}

if for all

\beta \in B

there exists

\alpha \in A

such that

V_{\beta }\subseteq U_{\alpha }.

In other words, there is a refinement map $\phi :B\to A$ 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 on the set of covers of $X$ is transitive and reflexive, i.e. a Preorder. It is never asymmetric for $X\neq \emptyset$ .

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 $a_{0}<a_{1}<\cdots <a_{n}$ being $a_{0}<b_{0}<a_{1}<a_{2}<\cdots <a_{n-1}<b_{1}<a_{n}$ ), 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 ${\mathcal {A}}=\{A\in {\mathcal {B}}:{\text{ there exists }}U\in {\mathcal {O}}{\text{ such that }}A\subseteq U\}.$ 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 ${\mathcal {C}}=\{U_{A}\in {\mathcal {O}}:A\in {\mathcal {A}}\}$ 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.

[1]

[2]