Covers are commonly used in the context of topology. If the set X is a topological space, then a coverC of X is a collection of subsets Uα of X whose union is the whole space X. In this case we say that CcoversX, or that the sets UαcoverX. 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 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 if each of its members is an open set. 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 = is locallyfinite if for any x ∈ X, there exists some neighborhood N of x such that the set 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, In other words, there is a refinement map satisfying for every. 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, considering topologies. When subdividing simplicial complexes, 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 be a topological basis of and be an open cover of. First take. Then is a refinement of. Next, for each, we select a containing . Then is a subcover of. 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, ; ;Lindelöf: if every open cover has a countable subcover, ; ;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 dimensionn 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.
