Cover (topology)
In mathematics, and more particularly in set theory, a cover (or covering)[1] of a setis an indexed family of subsets(indexed by the setCovers are commonly used in the context of topology.is a topological space, then a coveritself or sets in the parent spaceis said to be locally finite if every point ofhas a neighborhood that intersects only finitely many sets in the cover.is contained in only finitely many sets in the cover.[1] A cover is point finite if locally finite, though the converse is not necessarily true.be a cover of a topological spaceis said to be an open cover if each of its members is an open set.[1] A simple way to get a subcover is to omit the sets contained in another set in the cover.Consider specifically open covers.(requiring the axiom of choice).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.Formally, In other words, there is a refinement mapThis map is used, for instance, in the Čech cohomology of[2] 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 ofis transitive and reflexive, i.e. a Preorder.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), 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.The language of covers is often used to define several topological properties related to compactness.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.[3] If no such minimal n exists, the space is said to be of infinite covering dimension.