# Suslin condition

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.

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.