Cover (topology)

Cover (topology)

Cover in 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 Uα of X whose union is the whole space X. In this case we say that C covers X, or that the sets Uα cover X. 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 open cover 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 which intersects only finitely many sets in the cover. Formally, C = {Uα} is locally finite if for any x ∈ X, 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 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,
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.
Yet another notion of refinement is that of star refinement.
Subcover
Compactness
The language of covers is often used to define several topological properties related to compactness. A topological space X is said to be
- Compactif every open cover has a finite subcover, (or equivalently that every open cover has a finite refinement);Lindelöfif every open cover has acountablesubcover, (or equivalently that every open cover has a countable refinement);Metacompactif every open cover has a point finite open refinement;Paracompactif 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.[2] If no such minimal n exists, the space is said to be of infinite covering dimension.
See also
Atlas (topology)
Covering space
Partition of a set
Set cover problem