View source for 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 (mathematics)|set]] <math>X</math> is a [[Family of sets|family]] of [[subset]]s of <math>X</math> whose union is all of <math>X</math>. More formally, if <math>C = \lbrace U_\alpha : \alpha \in A \rbrace</math> is an [[indexed family]] of subsets <math>U_\alpha\subset X</math> (indexed by the set <math>A</math>), then <math>C</math> is a cover of <math>X</math> if <math>\bigcup_{\alpha \in A}U_{\alpha} \supseteq X</math>. Thus the collection <math>\lbrace U_\alpha : \alpha \in A \rbrace</math> is a cover of <math>X</math> if each element of <math>X</math> belongs to at least one of the subsets <math>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>X</math> is a [[topological space]], then a ''cover'' <math>C</math> of <math>X</math> is a collection of subsets <math>\{U_\alpha\}_{\alpha\in A}</math> of <math>X</math> whose union is the whole space <math>X</math>. In this case we say that <math>C</math> ''covers'' <math>X</math>, or that the sets <math>U_\alpha</math> ''cover'' <math>X</math>. 

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

:<math>Y \subseteq \bigcup_{\alpha \in A}U_{\alpha}.</math>

That is, we may cover <math>Y</math> with either sets in <math>Y</math> itself or sets in the parent space <math>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 '''{{vanchor|open cover}}''' if each of its members is an [[open set]] (i.e. each ''U''<sub>''α''</sub> is contained in ''T'', where ''T'' is the topology on ''X'').

A cover of ''X'' is said to be [[locally finite collection|locally finite]] if every point of ''X'' has a [[neighborhood (topology)|neighborhood]] that intersects only [[finite set|finite]]ly many sets in the cover.  Formally, ''C'' = {''U''<sub>''α''</sub>} is locally finite if for any <math>x \in X,</math> there exists some neighborhood ''N''(''x'') of ''x'' such that the set

:<math>\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>C</math> of a topological space <math>X</math> is a new cover <math>D</math> of <math>X</math> such that every set in <math>D</math> is contained in some set in <math>C</math>. Formally,

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

In other words, there is a '''refinement map''' <math>\phi : B \to A</math> satisfying <math>V_{\beta} \subseteq U_{\phi(\beta)}</math> for every <math>\beta \in B.</math> This map is used, for instance, in the  [[Čech cohomology]] of <math>X</math>.<ref>{{Cite book|title = Differential Forms in Algebraic Topology|date = 1982|last = Bott, Tu|page = 111}}</ref>

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>X</math> is [[transitive relation|transitive]] and [[reflexive relation|reflexive]], i.e. a [[Preorder]]. It is never [[symmetric relation|asymmetric]] for <math>X\neq\empty</math>.

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 (mathematics)|interval]] (one refinement of <math>a_0 < a_1 < \cdots < a_n</math> being <math>a_0 < b_0 < a_1 < a_2 < \cdots < a_{n-1} < b_1 < a_n</math>), considering [[topology|topologies]] (the [[standard topology]] in Euclidean space being a refinement of the [[trivial topology]]). When subdividing [[simplicial complex]]es (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>\mathcal{B}</math> be a topological basis of <math>X</math> and <math>\mathcal{O}</math> be an open cover of <math>X.</math> First take <math>\mathcal{A} = \{ A \in \mathcal{B} : \text{ there exists } U \in \mathcal{O} \text{ such that } A \subseteq U \}.</math> Then <math>\mathcal{A}</math> is a refinement of <math>\mathcal{O}</math>. Next, for each <math>A \in \mathcal{A},</math> we select a <math>U_{A} \in \mathcal{O}</math> containing <math>A</math> (requiring the axiom of choice). Then <math>\mathcal{C} = \{ U_{A} \in \mathcal{O} : A \in \mathcal{A} \}</math> is a subcover of <math>\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 space|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 space|Compact]]: if every open cover has a finite subcover, (or equivalently that every open cover has a finite refinement);
;[[Lindelöf space|Lindelöf]]: if every open cover has a [[countable]] subcover, (or equivalently that every open cover has a countable refinement);
;[[metacompact space|Metacompact]]: if every open cover has a point-finite open refinement;
;[[paracompact space|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.<ref>{{cite book
 | first = James
 | last = Munkres
 | author-link = James Munkres
 | year = 1999
 | title = Topology
 | edition = 2nd
 | publisher = [[Prentice Hall]]
 | isbn = 0-13-181629-2
}}</ref> If no such minimal ''n'' exists, the space is said to be of infinite covering dimension.

==See also==

* {{annotated link|Atlas (topology)}}
* {{annotated link|Bornology}}
* {{annotated link|Covering space}}
* {{annotated link|Grothendieck topology}}
* {{annotated link|Partition of a set}}
* {{annotated link|Set cover problem}}
* {{annotated link|Star refinement}}
* {{annotated link|Subpaving}}

==Notes==
{{reflist}}
{{reflist|group=note}}

==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==
* {{springer|title=Covering (of a set)|id=p/c026950}}

[[Category:Topology]]
[[Category:General topology]]
[[Category:Families of sets]]