Cover (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 = \bigcup_{\alpha \in A}U_{\alpha}. In this case C is said to cover X, or that the sets U_\alpha cover X. 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. That is, C is a cover of Y if Y \subseteq \bigcup_{\alpha \in A}U_{\alpha}. Here, Y may be covered with either sets in Y itself or sets in the parent space 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 = \{U_\alpha\} 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 locally finite, though the converse is not necessarily true. == Subcover ==

Let C be a cover of a topological space X . A subcover of C is a subset of C that still covers X . The cover C is said to be an '''' if each of its members is an open set. That is, each U_\alpha is contained in T , where T is the topology on X. 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}, one may 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, second countability implies space is Lindelöf. == 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 and reflexive, i.e. a Preorder. It is never asymmetric for X\neq\empty. 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. ==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; and • orthocompact: if every open cover has an interior-preserving 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. ==See also==