# Cover (topology)

In mathematics, particularly topology, a cover of a set $X$ is a collection of sets whose union includes $X$ as a subset. Formally, if $C=\lbrace U_{\alpha }:\alpha \in A\rbrace$ is an indexed family of sets $U_{\alpha },$ then $C$ is a cover of $X$ if

$X\subseteq \bigcup _{\alpha \in A}U_{\alpha }.$ ## Cover in topology

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

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

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

That is, we may cover $Y$  with either open sets in $Y$  itself, or cover $Y$  by open sets in the parent space $X$ .

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

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

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 $C$  of a topological space $X$  is a new cover $D$  of $X$  such that every set in $D$  is contained in some set in $C$ . Formally,

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

In other words, there is a refinement map $\phi :B\to A$  satisfying $V_{\beta }\subseteq U_{\phi (\beta )}$  for every $\beta \in B.$  This map is used, for instance, in the Čech cohomology of $X$ .

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 is a preorder on the set of covers of $X$ .

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}  being $a_{0} ), 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 ${\mathcal {B}}$  be a topological basis of $X$  and ${\mathcal {O}}$  be an open cover of $X.$  First take ${\mathcal {A}}=\{A\in {\mathcal {B}}:{\text{ there exists }}U\in {\mathcal {O}}{\text{ such that }}A\subseteq U\}.$  Then ${\mathcal {A}}$  is a refinement of ${\mathcal {O}}$ . Next, for each $A\in {\mathcal {A}},$  we select a $U_{A}\in {\mathcal {O}}$  containing $A$  (requiring the axiom of choice). Then ${\mathcal {C}}=\{U_{A}\in {\mathcal {O}}:A\in {\mathcal {A}}\}$  is a subcover of ${\mathcal {O}}.$  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. If no such minimal n exists, the space is said to be of infinite covering dimension.