Cover (topology)

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

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.

Covers are commonly used in the context of topology. If the set ${\displaystyle X}$ is a topological space, then a cover ${\displaystyle C}$ of ${\displaystyle X}$ is a collection of subsets ${\displaystyle \{U_{\alpha }\}_{\alpha \in A}}$ of ${\displaystyle X}$ whose union is the whole space ${\displaystyle X}$. In this case we say that ${\displaystyle C}$ covers ${\displaystyle X}$, or that the sets ${\displaystyle U_{\alpha }}$ cover ${\displaystyle X}$.

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

${\displaystyle Y\subseteq \bigcup _{\alpha \in A}U_{\alpha }.}$

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

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

${\displaystyle \left\{\alpha \in A:U_{\alpha }\cap N(x)\neq \varnothing \right\}}$

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.

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

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

In other words, there is a refinement map ${\displaystyle \phi :B\to A}$ satisfying ${\displaystyle V_{\beta }\subseteq U_{\phi (\beta )}}$ for every ${\displaystyle \beta \in B.}$ This map is used, for instance, in the Čech cohomology of ${\displaystyle X}$.^[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 ${\displaystyle X}$ 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 ${\displaystyle a_{0}<a_{1}<\cdots <a_{n}}$ being ${\displaystyle a_{0}<b_{0}<a_{1}<a_{2}<\cdots <a_{n-1}<b_{1}<a_{n}}$), 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.

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 ${\displaystyle X}$ and ${\displaystyle {\mathcal {O}}}$ be an open cover of ${\displaystyle X.}$ First take ${\displaystyle {\mathcal {A}}=\{A\in {\mathcal {B}}:{\text{ there exists }}U\in {\mathcal {O}}{\text{ such that }}A\subseteq U\}.}$ Then ${\displaystyle {\mathcal {A}}}$ is a refinement of ${\displaystyle {\mathcal {O}}}$. Next, for each ${\displaystyle A\in {\mathcal {A}},}$ we select a ${\displaystyle U_{A}\in {\mathcal {O}}}$ containing ${\displaystyle A}$ (requiring the axiom of choice). Then ${\displaystyle {\mathcal {C}}=\{U_{A}\in {\mathcal {O}}:A\in {\mathcal {A}}\}}$ is a subcover of ${\displaystyle {\mathcal {O}}.}$ 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.

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.

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.

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