{{Short description|Subsets whose union equals the whole set}} {{redirect|Cover (mathematics)||Cover (disambiguation)#Mathematics}} {{about|a family of subsets whose union is the whole set|a type of continuous map|covering space}}
In mathematics, and more particularly in set theory, a '''cover''' (or '''covering'''){{r|willard}} of a set <math>X</math> is a family of subsets 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 display="block"> \bigcup_{\alpha \in A}U_{\alpha} = 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>.
== Definition == 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 = \bigcup_{\alpha \in A}U_{\alpha}</math>. In this case <math> C </math> is said to cover <math>X</math>, or that the sets <math>U_\alpha</math> cover <math>X</math>.{{r|willard}}
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>. That is, <math> C</math> is a cover of <math>Y</math> if <math display="block">Y \subseteq \bigcup_{\alpha \in A}U_{\alpha}.</math> Here, <math>Y</math> may be covered with either sets in <math>Y</math> itself or sets in the parent space <math>X</math>.
A cover of <math> X </math> is said to be locally finite if every point of <math> X </math> has a neighborhood that intersects only finitely many sets in the cover. Formally, <math> C = \{U_\alpha\} </math> is locally finite if, for any <math>x \in X</math>, there exists some neighborhood <math> N(x) </math> of <math> x </math> such that the set <math display="block"> \left\{ \alpha \in A : U_{\alpha} \cap N(x) \neq \varnothing \right\}</math> is finite. A cover of <math> X </math> is said to be ''point finite'' if every point of <math> X </math> is contained in only finitely many sets in the cover.{{r|willard}} A cover is point finite if locally finite, though the converse is not necessarily true.
== Subcover == Let <math> C </math> be a cover of a topological space <math> X </math>. A ''subcover'' of <math> C </math> is a subset of <math> C </math> that still covers <math> X </math>. The cover <math> C </math> is said to be an ''{{vanchor|open cover}}'' if each of its members is an open set. That is, each <math> U_\alpha </math> is contained in <math> T </math>, where <math> T </math> is the topology on <math>X</math>.{{r|willard}}
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> one may 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, second countability implies space is Lindelöf.
== 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>.{{r|bott}}
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 and reflexive, i.e. a Preorder. It is never 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 (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 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.
==Compactness== The language of covers is often used to define several topological properties related to compactness. A topological space <math> X </math> 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 <math>X</math> is said to be of covering dimension ''<math>n</math>'' if every open cover of <math>X</math> has a point-finite open refinement such that no point of <math>X</math> is included in more than ''<math>n+1</math>'' sets in the refinement and if ''<math>n</math>'' is the minimum value for which this is true.{{r|munkres}} If no such minimal ''<math>n</math>'' 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}}
==References== <references> <ref name=bott>{{cite book | last = Bott | first = Tu | title = Differential Forms in Algebraic Topology | year = 1982 | page = 111 }}</ref>
<ref name="munkres">{{cite book | last = Munkres | first = James | author-link = James Munkres | year = 1999 | title = Topology | edition = 2nd | publisher = Prentice Hall | isbn = 0-13-181629-2 }}</ref>
<ref name="willard">{{cite book | last = Willard | first = Stephen | title = General Topology | year = 1998 | publisher = Dover Publications | url = https://books.google.com/books?id=-o8xJQ7Ag2cC&pg=PA104 | page = 104 | isbn = 0-486-43479-6 }}</ref> </references>
* ''Introduction to Topology'', Second Edition, Theodore W. Gamelin & Robert Everist Greene. Dover Publications 1999. {{ISBN|0-486-40680-6}} * {{Kelley 1975}}
==External links== * {{springer|title=Covering (of a set)|id=p/c026950}}
Category:Topology Category:General topology Category:Families of sets