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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021290/c0212901.png" /> of subsets of the series of natural numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021290/c0212902.png" /> of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021290/c0212903.png" /> 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 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.
  
 
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.
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021290/c0212904.png" /> be an infinite cardinal number. Then an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021290/c0212906.png" />-centred family of sets is defined as a family for which the intersection of any set of elements of cardinality less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021290/c0212907.png" /> is not empty. Such families are used to characterize <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021290/c0212908.png" />-compact spaces in abstract measure theory.
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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====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.L. Kelley,  "General topology" , Springer  (1975)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L. Gillman,  M. Jerison,  "Rings of continuous functions" , v. Nostrand-Reinhold  (1960)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  J.L. Kelley,  "General topology" , Springer  (1975)</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  L. Gillman,  M. Jerison,  "Rings of continuous functions" , v. Nostrand-Reinhold  (1960)</TD></TR>
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</table>
  
  
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====Comments====
 
====Comments====
 
A centred family of sets is also called a filtered family of sets or simply a filter.
 
A centred family of sets is also called a filtered family of sets or simply a filter.
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Revision as of 21:38, 17 December 2015

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.

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 $\mathfrak{m}$ 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=13084
This article was adapted from an original article by B.A. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article