Cover (topology)

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

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 $$X$$ 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 sets in $$Y$$ itself or 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 on the set of covers of $$X$$ 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 $$a_0 < a_1 < \cdots < a_n$$ being $$a_0 < b_0 < a_1 < a_2 < \cdots < a_{n-1} < b_1 < a_n$$), 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.

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.