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In this paper we introduce some fusion properties of forcing notions which guarantee that an iteration with supports of size ≤ κ not only does not collapse κ but also preserves the strength of κ (after a suitable preparatory forcing). This provides a general theory covering the known cases of tree iterations which preserve large cardinals (cf. [4, 5, 6, 8,… (More)
Using a dictionary translating a variety of classical and modern covering properties into combinatorial properties of continuous images, we get a simple way to understand the interrelations between these properties in ZFC and in the realm of the trichotomy axiom for upward closed families of sets of natural numbers. While it is now known that the answer to… (More)
We show that b = c = ω3 is consistent with the existence of a ∆3-definable wellorder of the reals and a Π2-definable ω-mad subfamily of [ω] ω (resp. ω).
In this paper we characterize various sorts of boundedness of the free (abelian) topological group F (X) (A(X)) as well as the free locally-convex linear topological space L(X) in terms of properties of a Tychonoff space X. These properties appear to be close to so-called selection principles, which permits us to show, that (it is consistent with ZFC that)… (More)
Assuming the absence of Q-points (which is consistent with ZFC) we prove that the free topological group F (X) over a Tychonov space X is o-bounded if and only if every continuous metrizable image T of X satisfies the selection principle S fin(O,Ω) (the latter means that for every sequence 〈un〉n∈ω of open covers of T there exists a sequence 〈vn〉n∈ω such… (More)
We show that even for subsetsX of the real line which do not contain perfect sets, the Hurewicz property does not imply the property S1(Γ,Γ), asserting that for each countable family of open γ-covers of X , there is a choice function whose image is a γcover of X . This settles a problem of Just, Miller, Scheepers, and Szeptycki. Our main result also answers… (More)
We prove that if an ultrafilter L is not coherent to a Q-point, then each analytic non-σ-bounded topological group G admits an increasing chain 〈Gα : α < b(L)〉 of its proper subgroups such that: (i) ⋃ α Gα = G; and (ii) For every σ-bounded subgroup H of G there exists α such that H ⊂ Gα. In case of the group Sym(ω) of all permutations of ω with the topology… (More)
We extend the work of  by presenting a method for controlling cardinal characteristics in the presence of a projective wellorder and 2א0 > א2. This also answers a question of Harrington  by showing that the existence of a ∆3 wellorder of the reals is consistent with Martin’s axiom and 2א0 = א3.