Sacks reals and Martin's axiom
Selection from Upper Semi-Continuous Compact-Valued Mappings.
Sequential completeness and -sequential completeness are different
Set mappings on generalized linear continua
Sets with no uncountable Blackwell subsets
Set-theoretic constructions of two-point sets
A two-point set is a subset of the plane which meets every line in exactly two points. By working in models of set theory other than ZFC, we demonstrate two new constructions of two-point sets. Our first construction shows that in ZFC + CH there exist two-point sets which are contained within the union of a countable collection of concentric circles. Our second construction shows that in certain models of ZF, we can show the existence of two-point sets without explicitly invoking the Axiom of Choice....
Small almost free modules with prescribed topological endomorphism rings
Some additive properties of special sets of reals
Some Baire category type theorems for
Some new versions of an old game
The old game is the point-open one discovered independently by F. Galvin [7] and R. Telgársky [17]. Recall that it is played on a topological space as follows: at the -th move the first player picks a point and the second responds with choosing an open . The game stops after moves and the first player wins if . Otherwise the victory is ascribed to the second player. In this paper we introduce and study the games and . In the moves are made exactly as in the point-open game, but the...
Some remarks on embeddings of Boolean algebras and topological spaces
Some remarks on generalized Martin's Axiom.
Some remarks on subgroups of real numbers
Squares with diamonds and Souslin trees with special squares
Strongly almost disjoint familes, revisited
The relations M(κ,λ,μ) → B [resp. B(σ)] meaning that if with |A|=κ is μ-almost disjoint then A has property B [resp. has a σ-transversal] had been introduced and studied under GCH in [EH]. Our two main results here say the following: Assume GCH and let ϱ be any regular cardinal with a supercompact [resp. 2-huge] cardinal above ϱ. Then there is a ϱ-closed forcing P such that, in , we have both GCH and [resp. for all . These show that, consistently, the results of [EH] are sharp. The necessity...
Subgroups of the Baer–Specker group with few endomorphisms but large dual
Assuming the continuum hypothesis, we construct a pure subgroup G of the Baer-Specker group with the following properties. Every endomorphism of G differs from a scalar multiplication by an endomorphism of finite rank. Yet G has uncountably many homomorphisms to ℤ.
Supercompactness and failures of GCH
Let κ < λ be regular cardinals. We say that an embedding j: V → M with critical point κ is λ-tall if λ < j(κ) and M is closed under κ-sequences in V. Silver showed that GCH can fail at a measurable cardinal κ, starting with κ being κ⁺⁺-supercompact. Later, Woodin improved this result, starting from the optimal hypothesis of a κ⁺⁺-tall measurable cardinal κ. Now more generally, suppose that κ ≤ λ are regular and one wishes the GCH to fail at λ with κ being λ-supercompact. Silver’s methods show...
Sur les mesures vectorielles définies par une application Pettis-intégrable