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About cover In mathematics, and more particularly in set theory, a cover of a set is a family of subsets of whose union is all of . More formally, if is an indexed family of subsets , then is a cover of if
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Article 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 }}
.
Contents 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 }}
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
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
=
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.
Article from Wikipedia (CC BY-SA 4.0), where it is maintained by volunteer editors.
{\displaystyle U_{\alpha }}
is a (topological) subspace of
is a collection of subsets
{\displaystyle C=\{U_{\alpha }\}_{\alpha \in A}}
{\displaystyle Y\subseteq \bigcup _{\alpha \in A}U_{\alpha }.}
may be covered with either sets in
itself or sets in the parent space
is said to be locally finite if every point of
has a neighborhood that intersects only finitely many sets in the cover. Formally,
{\displaystyle C=\{U_{\alpha }\}}
is locally finite if, for any
, there exists some neighborhood
{\displaystyle \left\{\alpha \in A:U_{\alpha }\cap N(x)\neq \varnothing \right\}}
is said to be point finite if every point of
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.
A simple way to get a subcover is to omit the sets contained in another set in the cover. Consider specifically open covers. Let
{\displaystyle {\mathcal {B}}}
be a topological basis of
{\displaystyle {\mathcal {O}}}
{\displaystyle {\mathcal {A}}=\{A\in {\mathcal {B}}:{\text{ there exists }}U\in {\mathcal {O}}{\text{ such that }}A\subseteq U\}}
{\displaystyle {\mathcal {A}}}
{\displaystyle {\mathcal {O}}}
{\displaystyle A\in {\mathcal {A}},}
{\displaystyle U_{A}\in {\mathcal {O}}}
(requiring the axiom of choice). Then
{\displaystyle {\mathcal {C}}=\{U_{A}\in {\mathcal {O}}:A\in {\mathcal {A}}\}}
{\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.
{\displaystyle C=\{U_{\alpha }\}_{\alpha \in A}}
{\displaystyle \beta \in B}
{\displaystyle \alpha \in A}
{\displaystyle V_{\beta }\subseteq U_{\alpha }.}
In other words, there is a refinement map
{\displaystyle \phi :B\to A}
{\displaystyle V_{\beta }\subseteq U_{\phi (\beta )}}
{\displaystyle \beta \in B.}
This map is used, for instance, in the Čech cohomology of
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
is transitive and reflexive, i.e. a Preorder. It is never asymmetric for
{\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
{\displaystyle a_{0}<a_{1}<\cdots <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.
In mathematics, and more particularly in set theory, a cover of a set is a family of subsets of whose union is all of . More formally, if is an indexed family of subsets , then is a cover of if
