Abstract We prove that a number of axioms, each a consequence of PFA (the Proper Forcing Axiom) are equivalent. In particular we show that TOP (the Thinning-out Principle as introduced by Baumgartner in the Handbook of set-theoretic topology), is equivalent to the following statement: If I is an ideal on ω1 with ω1 generators, then there exists an uncountable X ⊆ ω1, such that either [X]ω ∩ I = ∅ or [X]ω ⊆ I.

Abstract We discuss saturating ultrafilters on N, relating them to other types of non-principal ultrafilter. (a) There is an (ω,c)-saturating ultrafllter on N iff 2λ≤ c for everyλ