Coverage (mathematics)
In mathematics, coverage is a fundamental concept from set theory . Open coverings play an important role in the compactness of topological spaces .
Definitions
Overlap
A family of subsets of is called covering of if
applies. The coverage is called finite (or countable ) if the index set is finite (or countable).
Partial coverage
Are and covers of , then is called partial cover of , if for each one exists with .
refinement
If and are two overlaps of , then is called finer than if there is an index for each such that it holds. The system of sets is then called refinement or refinement coverage of .
Quasi-shrinkage and shrinkage
A refinement, as defined above, is called a quasi-shrinkage, if even holds. Is in addition and for all , it is called a contraction.
Coverings in topological spaces
Open / closed coverage
A covering of a topological space is called open (or closed ) if all in are open (or closed).
compactness
A topological space is called compact if every open cover of contains a finite partial cover .
Overlap properties
- An overlap is called punctual if every point of the space lies in at most finitely many overlap sets. A topological space is called metacompact if every open cover has a point refinement.
- An overlap is called locally finite if every point of the space has a neighborhood that intersects at most a finite number of overlap sets. As is well known, a topological space is called paracompact if every open cover has a locally finite refinement.
- A cover is called -local finite , if it can be written as a countable union of families of sets , so that each point of space has a neighborhood for each , which cuts out at most finitely many sets .
- A cover is called -discreet if it can be written as a countable union of families of sets , so that for each point and for each there is a neighborhood of this point which cuts out at most one of the sets . The -discrete and -local coverages play an important role in the Bing-Nagata-Smirnov theorem .
normality
A T _{1} -space is normal if and only if every open locally finite cover has a shrinkage.
See also
literature
- Boto von Querenburg : Set theoretical topology (= Springer textbook ). 3rd, revised and expanded edition. Springer, Berlin et al. 2001, ISBN 3-540-67790-9
- Karl Peter Grotemeyer : Topology , Bibliographisches Institut Mannheim (1969), ISBN 3-411-00836-9