Centred family of sets

A family of sets with the finite intersection property: the intersection of any finite subfamily is non-empty. For example, the countable family $\{ A_i : i \in \mathbf{Z} \}$ of subsets of the series of natural numbers $\mathbf{Z}_+$ of the form $A_i = \{ n \in \mathbf{Z} : n > i \}$ is centred; any family in which the intersection of all members 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: a space is compact if and only if every centred family of closed sets has non-empty intersection. 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 $\mathfrak{m}$ be an infinite cardinal number. Then an $\mathfrak{m}$-centred family of sets is defined as a family for which the intersection of any set of elements of cardinality less than $\mathfrak{m}$ is not empty. Such families are used to characterize $\mathfrak{m}$-compact spaces in abstract measure theory.

References

[1]	J.L. Kelley, "General topology" , Springer (1975)
[2]	L. Gillman, M. Jerison, "Rings of continuous functions" , v. Nostrand-Reinhold (1960)

Comments

A centred family of sets is also called a filtered family of sets or simply a filter.