Some relatives of the Juhász Club Principle are introduced and studied in the presence of CH. In particular, it is shown that a slight strengthening of this principle implies the existence of a Suslin tree in the presence of CH.
We show that under ZFC, for every indecomposable ordinal α < ω₁, there exists a poset which is β-proper for every β < α but not α-proper. It is also shown that a poset is forcing equivalent to a poset satisfying Axiom A if and only if it is α-proper for every α < ω₁.
Assuming V = L, for every successor cardinal κ we construct a GCH and cardinal preserving forcing poset ℙ ∈ L such that in the ideal of all non-stationary subsets of κ is Δ₁-definable over H(κ⁺).
We define -directedness, investigate various properties to determine whether they have this property or not, and use our results to obtain easier proofs of theorems due to Laurence and Alster concerning the existence of a Michael space, i.eȧ Lindelöf space whose product with the irrationals is not Lindelöf.
We show the property “is proper and preserves every -Souslin tree” is preserved by countable support iteration.