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We present principles for guessing clubs in the generalized club filter on P κ λ. These principles are shown to be weaker than classical diamond principles but often serve as sufficient substitutes. One application is a new construction of a λ +-Suslin-tree using assumptions different from previous constructions. The other application partly solves open… (More)
We show that large fragments of MM, e. g. the tree property and stationary reflection, are preserved by strongly (ω1 + 1)-game-closed forcings. PFA can be destroyed by a strongly (ω1 + 1)-game-closed forcing but not by an ω2-closed.
It is known that the Stone-ˇ Cech compactification βX of a non-compact metrizable space X is approximated by the collection of Smirnov compactifications of X for all compatible metrics on X. We investigate the smallest cardinality of a set D of compatible metrics on the countable discrete space ω such that, βω is approximated by Smirnov compactifications… (More)
We show that for any infinite cardinal κ, every (κ+1)-strategically closed poset is κ +-strategically closed if and only if κ holds. This extends previous results of Velleman, et.al.