# 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 $\displaystyle{ X }$ 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 $\displaystyle{ \bigcup_{\alpha \in A}U_{\alpha} = X }$. 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.

## Cover in topology

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 Uα 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 = {Uα} 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.

## Refinement

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, irreflexive, and asymmetric.

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 $\displaystyle{ a_0 \lt a_1 \lt \cdots \lt a_n }$ being $\displaystyle{ a_0 \lt b_0 \lt a_1 \lt a_2 \lt \cdots \lt a_{n-1} \lt b_1 \lt a_n }$), 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.

## 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 $\displaystyle{ \mathcal{B} }$ 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.

## 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
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.

## 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.[2] If no such minimal n exists, the space is said to be of infinite covering dimension.