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Difference between revisions of "Centred family of sets"

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A family in which the intersection of any finite set of elements 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 elements is not empty is centred. Every finite centred family of sets has this last-named property.
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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. Centred families of closed sets in a topological space are used for the construction of its compactification and its absolute.
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Infinite centred families of sets were first used in [[general topology]] to characterize [[compact space]]s: 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.
 
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.

Latest revision as of 21:48, 17 December 2015

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.

How to Cite This Entry:
Centred family of sets. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Centred_family_of_sets&oldid=36966
This article was adapted from an original article by B.A. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article