Partition numbers Original Research Article

We continue [21] and study partition numbers of partial orderings which are related to ℘(ω)/fin. In particular, we investigate Pf, be the suborder of (℘(ω)/fin)ω containing only filtered elements, the Mathias partial order M, and (ω), (ω)ω the lattice of (infinite) partitions of ω, respectively. We show that Solomonʹs inequality holds for M and that it consistently fails for Pf. We show that the partition number of (ω) is C. We also show that consistently the distributivity number of (ω)ω is smaller than the distributivity number of ℘(ω)/fin. We also investigate partitions of a Polish space into closed sets. We show that such a partition either is countable or has size at least D, where D is the dominating number. We also show that the existence of a dominating family of size View the MathML source does not imply that a Polish space can be partitioned into View the MathML source many closed sets.