Cover (Topology)

Short description: Subsets whose union equals the whole set

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

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 [math]\displaystyle{ X }[/math] is a topological space, then a cover [math]\displaystyle{ C }[/math] of [math]\displaystyle{ X }[/math] is a collection of subsets [math]\displaystyle{ \{U_\alpha\}_{\alpha\in A} }[/math] of [math]\displaystyle{ X }[/math] whose union is the whole space [math]\displaystyle{ X }[/math]. In this case we say that [math]\displaystyle{ C }[/math] covers [math]\displaystyle{ X }[/math], or that the sets [math]\displaystyle{ U_\alpha }[/math] cover [math]\displaystyle{ X }[/math].

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

[math]\displaystyle{ Y \subseteq \bigcup_{\alpha \in A}U_{\alpha}. }[/math]

That is, we may cover [math]\displaystyle{ Y }[/math] with either sets in [math]\displaystyle{ Y }[/math] itself or sets in the parent space [math]\displaystyle{ X }[/math].

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 [math]\displaystyle{ x \in X, }[/math] there exists some neighborhood N(x) of x such that the set

[math]\displaystyle{ \left\{ \alpha \in A : U_{\alpha} \cap N(x) \neq \varnothing \right\} }[/math]

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.

Refinement

A refinement of a cover [math]\displaystyle{ C }[/math] of a topological space [math]\displaystyle{ X }[/math] is a new cover [math]\displaystyle{ D }[/math] of [math]\displaystyle{ X }[/math] such that every set in [math]\displaystyle{ D }[/math] is contained in some set in [math]\displaystyle{ C }[/math]. Formally,

[math]\displaystyle{ D = \{ V_{\beta} \}_{\beta \in B} }[/math] is a refinement of [math]\displaystyle{ C = \{ U_{\alpha} \}_{\alpha \in A} }[/math] if for all [math]\displaystyle{ \beta \in B }[/math] there exists [math]\displaystyle{ \alpha \in A }[/math] such that [math]\displaystyle{ V_{\beta} \subseteq U_{\alpha}. }[/math]

In other words, there is a refinement map [math]\displaystyle{ \phi : B \to A }[/math] satisfying [math]\displaystyle{ V_{\beta} \subseteq U_{\phi(\beta)} }[/math] for every [math]\displaystyle{ \beta \in B. }[/math] This map is used, for instance, in the Čech cohomology of [math]\displaystyle{ X }[/math].^[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 [math]\displaystyle{ X }[/math] is transitive, irreflexive, and asymmetric.

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 [math]\displaystyle{ a_0 \lt a_1 \lt \cdots \lt a_n }[/math] being [math]\displaystyle{ a_0 \lt b_0 \lt a_1 \lt a_2 \lt \cdots \lt a_{n-1} \lt b_1 \lt a_n }[/math]), 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 [math]\displaystyle{ \mathcal{B} }[/math] be a topological basis of [math]\displaystyle{ X }[/math] and [math]\displaystyle{ \mathcal{O} }[/math] be an open cover of [math]\displaystyle{ X. }[/math] First take [math]\displaystyle{ \mathcal{A} = \{ A \in \mathcal{B} : \text{ there exists } U \in \mathcal{O} \text{ such that } A \subseteq U \}. }[/math] Then [math]\displaystyle{ \mathcal{A} }[/math] is a refinement of [math]\displaystyle{ \mathcal{O} }[/math]. Next, for each [math]\displaystyle{ A \in \mathcal{A}, }[/math] we select a [math]\displaystyle{ U_{A} \in \mathcal{O} }[/math] containing [math]\displaystyle{ A }[/math] (requiring the axiom of choice). Then [math]\displaystyle{ \mathcal{C} = \{ U_{A} \in \mathcal{O} : A \in \mathcal{A} \} }[/math] is a subcover of [math]\displaystyle{ \mathcal{O}. }[/math] 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.