View source for 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.



====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====
<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>

[[Category:Order, lattices, ordered algebraic structures]]
[[Category:General topology]]