In mathematics, and more particularly in set theory, a cover (or covering)[1] of a set
is a family of subsets of
is an indexed family of subsets
Covers are commonly used in the context of topology.
is a topological space, then a cover
itself or sets in the parent space
is said to be locally finite if every point of
has 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 space
is 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 map
This 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 of
is 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.