Centred family of sets
A family in which the intersection of any finite set of elements is non-empty. For example, the countable family of subsets of the series of natural numbers of the form is centred; any family in which the intersection of all elements is not empty is centred. Every finite centred family of sets has this last-named property.
Infinite centred families of sets were first used in general topology to characterize compact spaces. Centred families of closed sets in a topological space are used for the construction of its compactification and its absolute.
The concept of a centred system of sets can be generalized as follows. Let be an infinite cardinal number. Then an -centred family of sets is defined as a family for which the intersection of any set of elements of cardinality less than is not empty. Such families are used to characterize -compact spaces in abstract measure theory.
|||J.L. Kelley, "General topology" , Springer (1975)|
|||L. Gillman, M. Jerison, "Rings of continuous functions" , v. Nostrand-Reinhold (1960)|
A centred family of sets is also called a filtered family of sets or simply a filter.
Centred family of sets. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Centred_family_of_sets&oldid=13084