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Difference between revisions of "Suslin condition"

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In partially ordered sets, the Suslin condition is commonly called the countable anti-chain condition. In Boolean algebras, it is equivalent to the assertion that every totally ordered subset is countable; for this reason it is often called the countable chain condition, and this usage is also (misleadingly) applied to partially ordered sets.
 
In partially ordered sets, the Suslin condition is commonly called the countable anti-chain condition. In Boolean algebras, it is equivalent to the assertion that every totally ordered subset is countable; for this reason it is often called the countable chain condition, and this usage is also (misleadingly) applied to partially ordered sets.
  
The Suslin number of a topological space $X$ is the minimum cardinal number $\kappa$ such that every pairwise disjoint family of open subsets of $X$ has cardinality less than $\kappa$. This is closely related to the cellularity: the supremum of the cardinalities of pairwise disjoint families of open subsets.
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The Suslin number of a topological space $X$ is the minimum cardinal number $\kappa$ such that every pairwise disjoint family of open subsets of $X$ has cardinality less than $\kappa$: cf. [[Cardinal characteristic]]. This is closely related to the cellularity: the supremum of the cardinalities of pairwise disjoint families of open subsets.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W.W. Comfort,  S. Negrepontis,  "Chain conditions in topology" , Cambridge Univ. Press  (1982)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W.W. Comfort,  S. Negrepontis,  "Chain conditions in topology" , Cambridge Univ. Press  (1982)</TD></TR></table>
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[[Category:Order, lattices, ordered algebraic structures]]
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[[Category:General topology]]

Latest revision as of 21:33, 9 November 2014

A condition that arose when the Suslin hypothesis was stated. A topological space (a Boolean algebra, a partially ordered set) satisfies the Suslin condition if and only if every family of non-empty disjoint open subsets (of non-zero pairwise incompatible elements) is countable. The Suslin condition has been generalized to include an arbitrary cardinal number; the corresponding cardinal-valued invariant is the Suslin number.


Comments

In partially ordered sets, the Suslin condition is commonly called the countable anti-chain condition. In Boolean algebras, it is equivalent to the assertion that every totally ordered subset is countable; for this reason it is often called the countable chain condition, and this usage is also (misleadingly) applied to partially ordered sets.

The Suslin number of a topological space $X$ is the minimum cardinal number $\kappa$ such that every pairwise disjoint family of open subsets of $X$ has cardinality less than $\kappa$: cf. Cardinal characteristic. This is closely related to the cellularity: the supremum of the cardinalities of pairwise disjoint families of open subsets.

References

[a1] W.W. Comfort, S. Negrepontis, "Chain conditions in topology" , Cambridge Univ. Press (1982)
How to Cite This Entry:
Suslin condition. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Suslin_condition&oldid=32794
This article was adapted from an original article by V.I. Malykhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article