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AbstractWe 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 theHandbook of set-theoretic topology), is equivalent to the following statement: IfIis an ideal onω1withω1generators, then there exists an uncountableX⊆ω1, such that either [X]ω∩I= ∅ or [X]ω⊆I.doi:10.2307/2275840 fatcat:enmqg43vsbgrvf4daq7rjdju3a