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

Definition

Covers are commonly used in the context of topology. If the set X {\displaystyle X} is a topological space, then a cover C {\displaystyle C} of X {\displaystyle X} is a collection of subsets { U α } α ∈ A {\displaystyle \{U_{\alpha }\}_{\alpha \in A}} of X {\displaystyle X} whose union is the whole space X = ⋃ α ∈ A U α {\displaystyle X=\bigcup _{\alpha \in A}U_{\alpha }}. In this case C {\displaystyle C} is said to cover X {\displaystyle X}, or that the sets U α {\displaystyle U_{\alpha }} cover X {\displaystyle X}.

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

A cover of X {\displaystyle X} is said to be locally finite if every point of X {\displaystyle X} has a neighborhood that intersects only finitely many sets in the cover. Formally, C = { U α } {\displaystyle C=\{U_{\alpha }\}} is locally finite if, for any x ∈ X {\displaystyle x\in X}, there exists some neighborhood N ( x ) {\displaystyle N(x)} of x {\displaystyle x} such that the set { α ∈ A : U α ∩ N ( x ) ≠ ∅ } {\displaystyle \left\{\alpha \in A:U_{\alpha }\cap N(x)\neq \varnothing \right\}} is finite. A cover of X {\displaystyle X} is said to be point finite if every point of X {\displaystyle 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 {\displaystyle C} be a cover of a topological space X {\displaystyle X}. A subcover of C {\displaystyle C} is a subset of C {\displaystyle C} that still covers X {\displaystyle X}. The cover C {\displaystyle C} is said to be an open cover if each of its members is an open set. That is, each U α {\displaystyle U_{\alpha }} is contained in T {\displaystyle T}, where T {\displaystyle T} is the topology on X {\displaystyle 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 {\displaystyle {\mathcal {B}}} be a topological basis of X {\displaystyle X} and O {\displaystyle {\mathcal {O}}} be an open cover of X {\displaystyle X}. First, take A = { A ∈ B : there exists U ∈ O such that A ⊆ U } {\displaystyle {\mathcal {A}}=\{A\in {\mathcal {B}}:{\text{ there exists }}U\in {\mathcal {O}}{\text{ such that }}A\subseteq U\}}. Then A {\displaystyle {\mathcal {A}}} is a refinement of O {\displaystyle {\mathcal {O}}}. Next, for each A ∈ A , {\displaystyle A\in {\mathcal {A}},} one may select a U A ∈ O {\displaystyle U_{A}\in {\mathcal {O}}} containing A {\displaystyle A} (requiring the axiom of choice). Then C = { U A ∈ O : A ∈ A } {\displaystyle {\mathcal {C}}=\{U_{A}\in {\mathcal {O}}:A\in {\mathcal {A}}\}} is a subcover of O . {\displaystyle {\mathcal {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.

Refinement

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

D = { V β } β ∈ B {\displaystyle D=\{V_{\beta }\}_{\beta \in B}} is a refinement of C = { U α } α ∈ A {\displaystyle C=\{U_{\alpha }\}_{\alpha \in A}} if for all β ∈ B {\displaystyle \beta \in B} there exists α ∈ A {\displaystyle \alpha \in A} such that V β ⊆ U α . {\displaystyle V_{\beta }\subseteq U_{\alpha }.}

In other words, there is a refinement map ϕ : B → A {\displaystyle \phi :B\to A} satisfying V β ⊆ U ϕ ( β ) {\displaystyle V_{\beta }\subseteq U_{\phi (\beta )}} for every β ∈ B . {\displaystyle \beta \in B.} This map is used, for instance, in the Čech cohomology of X {\displaystyle 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 {\displaystyle X} is transitive and reflexive, i.e. a Preorder. It is never asymmetric for X ≠ ∅ {\displaystyle 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 < ⋯ < a n {\displaystyle a_{0}<a_{1}<\cdots <a_{n}} being a 0 < b 0 < a 1 < a 2 < ⋯ < a n − 1 < b 1 < a n {\displaystyle 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.[citation needed]

Compactness

The language of covers is often used to define several topological properties related to compactness. A topological space X {\displaystyle 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 {\displaystyle X} is said to be of covering dimension n {\displaystyle n} if every open cover of X {\displaystyle X} has a point-finite open refinement such that no point of X {\displaystyle X} is included in more than n + 1 {\displaystyle n+1} sets in the refinement and if n {\displaystyle n} is the minimum value for which this is true. If no such minimal n {\displaystyle n} exists, the space is said to be of infinite covering dimension.

See also

External links