# Cover (topology)

(Redirected from Open cover)

In mathematics, a cover of a set $X$ is a collection of sets whose union contains $X$ 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 Uα of X whose union is the whole space X. In this case we say that C covers X, or that the sets Uα cover X. Also, if Y is a subset of X, then a cover of Y is a collection of subsets of X whose union contains Y, i.e., C is a cover of Y if

$Y \subseteq \bigcup_{\alpha \in A}U_{\alpha}$

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 which intersects only finitely many sets in the cover. Formally, C = {Uα} is locally finite if for any xX, 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. (locally finite implies point finite)

## 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,

$D = V_{\beta \in B}$

is a refinement of

$U_{\alpha \in A} \qquad \mbox{when} \qquad \forall \beta \ \exists \alpha \ V_\beta \subseteq U_\alpha$.

In other words, there is a refinement map $\phi: B \rightarrow A$ satisfying $V_{\beta} \subseteq U_{\phi(\beta)}$ for every $\beta \in B$. This map is used, for instance, in the Čech cohomology of X.[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 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 < a_1 < ... being $a_0 < b_0 < a_1 < a_2 < ... < a_n < b_1$), 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.

## 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.