Cover (topology)

Cover (topology)
In mathematics, and more particularly in set theory, a cover (or covering) of a set is a family of subsets of whose union is all of . More formally, if is an indexed family of subsets (indexed by the set ), then is a cover of if . Thus the collection is a cover of if each element of belongs to at least one of the subsets .
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 is a topological space, then a cover of is a collection of subsets of whose union is the whole space . In this case we say that covers , or that the sets cover .
Also, if is a (topological) subspace of , then a cover of is a collection of subsets of whose union contains , i.e., is a cover of if
That is, we may cover with either sets in itself or sets in the parent space .
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 (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 there exists some neighborhood N(x) 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 of a topological space is a new cover of such that every set in is contained in some set in . Formally,
is a refinement of if for all there exists such that
In other words, there is a refinement map satisfying for every This map is used, for instance, in the Čech cohomology of .
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, 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 being ), 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 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 (requiring the axiom of choice). 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, (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.
See also
Notes
References
Introduction to Topology, Second Edition, Theodore W. Gamelin & Robert Everist Greene. Dover Publications 1999.
General Topology, John L. Kelley. D. Van Nostrand Company, Inc. Princeton, NJ. 1955.
External links
Topology
General topology
Families of sets