Cover (topology)

In mathematics, a cover of a set X is a collection of sets such that X is a subset of the union of sets in the collection. In symbols, if
 * $$C = \lbrace U_\alpha: \alpha \in A\rbrace$$

is an indexed family of sets Uα, 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. In symbols, C = {Uα} is locally finite if for any x ∈ 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.

Refinement
A refinement of a cover C of X is a new cover D of X such that every set in D is contained in some set in C. In symbols,
 * $$D = V_{\beta \in B}$$

is a refinement of
 * $$U_{\alpha \in A} \qquad \mbox{when} \qquad \forall \beta \ \exists \alpha \ V_\beta \subseteq U_\alpha$$.

Every subcover is also a refinement, but not vice-versa. A subcover is made from the sets that are in the cover, but fewer 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.

Another useful notion of refinement is 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 For some more variations see the above articles.
 * compact if every open cover has a finite subcover. This is equivalent to the requirement that every open cover have a finite refinement.
 * Lindelöf if every open cover has a countable subcover. This is equivalent to the requirement 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.