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 $C = {U_{α} : α \in A}$ is an indexed family of subsets $U_{α} \subset X$ (indexed by the set $A$ ), then $C$ is a cover of $X$ if $⋃_{α \in A} U_{α} = X .$ Thus the collection ${U_{α} : α \in A}$ is a cover of $X$ if each element of $X$ belongs to at least one of the subsets $U_{α}$ .

Definition

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 ${U_{α}}_{α \in A}$ of $X$ whose union is the whole space $X = ⋃_{α \in A} U_{α}$ . In this case $C$ is said to cover $X$ , or that the sets $U_{α}$ cover $X$ .

If $Y$ is a (topological) subspace of $X$ , then a cover of $Y$ is a collection of subsets $C = {U_{α}}_{α \in A}$ of $X$ whose union contains $Y$ . That is, $C$ is a cover of $Y$ if $Y \subseteq ⋃_{α \in A} U_{α} .$ Here, $Y$ may be covered with either sets in $Y$ itself or sets in the parent space $X$ .

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 ${α \in A : U_{α} \cap N (x) \neq \emptyset}$ 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 locally finite, though the converse is not necessarily true.

Subcover

Let $C$ be a cover of a topological space $X$ . A subcover of $C$ is a subset of $C$ that still covers $X$ . The cover $C$ is said to be an open cover if each of its members is an open set. That is, each $U_{α}$ is contained in $T$ , where $T$ is the topology on $X$ .

A simple way to get a subcover is to omit the sets contained in another set in the cover. Consider specifically open covers. Let $ℬ$ be a topological basis of $X$ and $𝒪$ be an open cover of $X$ . First, take $𝒜 = {A \in ℬ : there exists U \in 𝒪 such that A \subseteq U}$ . Then $𝒜$ is a refinement of $𝒪$ . Next, for each $A \in 𝒜,$ one may select a $U_{A} \in 𝒪$ containing $A$ (requiring the axiom of choice). Then $𝒞 = {U_{A} \in 𝒪 : A \in 𝒜}$ is a subcover of $𝒪 .$ Hence the cardinality of a subcover of an open cover can be as small as that of any topological basis. Hence, second countability implies space is Lindelöf.

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_{β}}_{β \in B}

is a refinement of

C = {U_{α}}_{α \in A}

if for all

β \in B

there exists

α \in A

such that

V_{β} \subseteq U_{α} .

In other words, there is a refinement map $ϕ : B \to A$ satisfying $V_{β} \subseteq U_{ϕ (β)}$ for every $β \in B .$ This map is used, for instance, in the Čech cohomology of $X$ .

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 $X \neq ∅︀$ .

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 $a_{0} < a_{1} < \dots < a_{n}$ being $a_{0} < b_{0} < a_{1} < a_{2} < \dots < a_{n - 1} < b_{1} < a_{n}$ ), 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.^{[citation needed]}

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; and
orthocompact: if every open cover has an interior-preserving 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. If no such minimal $n$ exists, the space is said to be of infinite covering dimension.

References

1 2 3 4 Willard, Stephen (1998). General Topology. Dover Publications. p. 104. ISBN 0-486-43479-6.
↑ Bott, Tu (1982). Differential Forms in Algebraic Topology. p. 111.
↑ Munkres, James (1999). Topology (2nd ed.). Prentice Hall. ISBN 0-13-181629-2.

Introduction to Topology, Second Edition, Theodore W. Gamelin & Robert Everist Greene. Dover Publications 1999. ISBN 0-486-40680-6
Kelley, John L. (1975) [1955]. General Topology. Graduate Texts in Mathematics. Vol. 27 (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-90125-1. OCLC 1365153.

External links

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

[willard-1] 1 2 3 4 Willard, Stephen (1998). General Topology. Dover Publications. p. 104. ISBN 0-486-43479-6.

[bott-2] Bott, Tu (1982). Differential Forms in Algebraic Topology. p. 111.

[munkres-3] Munkres, James (1999). Topology (2nd ed.). Prentice Hall. ISBN 0-13-181629-2.