Namespaces
Variants
Actions

Difference between revisions of "Centred family of sets"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(Relationship to finite intersection property)
 
(One intermediate revision by the same user not shown)
Line 1: Line 1:
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.
+
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.
+
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 <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.
+
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>
+
<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>
  
  
Line 12: Line 15:
 
====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.
 +
 +
{{TEX|done}}

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=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