Difference between revisions of "Suslin condition"
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A condition that arose when the [[Suslin hypothesis|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. | A condition that arose when the [[Suslin hypothesis|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. | ||
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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 | + | 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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<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> | ||
Revision as of 11:44, 10 August 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 $\kappa$. 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) |
Suslin condition. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Suslin_condition&oldid=17195