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 $X$ as follows: at the $n$-th move the first player picks a point $x_n\in X$ and the second responds with choosing an open $U_n\ni x_n$. The game stops after $\omega$ moves and the first player wins if $\cup \{U_n:n\in \omega \}=X$. Otherwise the victory is ascribed to the second player. In this paper we introduce and study the games $\theta$ and $\Omega$. In $\theta$ the moves are made exactly as in the point-open game, but the...

The relations M(κ,λ,μ) → B [resp. B(σ)] meaning that if $A\subset {\left[\kappa \right]}^{\lambda }$ 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 $V^P$, we have both GCH and $M({\varrho }^{(+\varrho +1)},{\varrho }^{+},\varrho )\nrightarrow B$ [resp. $M({\varrho }^{(+\varrho +1)},\lambda ,\varrho )\nrightarrow B\left({\varrho }^{+}\right)$ for all $\lambda \le {\varrho }^{(+\varrho +1)}]$. These show that, consistently, the results of [EH] are sharp. The necessity...

Assuming the continuum hypothesis, we construct a pure subgroup G of the Baer-Specker group ${\mathbb{Z}}^{{\aleph }_0}$ 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.

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...

Using finite support iteration of ccc partial orders we provide a model of 𝔟 = κ < 𝔰 = κ⁺ for κ an arbitrary regular, uncountable cardinal.

Throughout this abstract, $G$ is a topological Abelian group and $\widehat{G}$ is the space of continuous homomorphisms from $G$ into the circle group $𝕋$ in the compact-open topology. A dense subgroup $D$ of $G$ is said to determine $G$ if the (necessarily continuous) surjective isomorphism $\widehat{G}\twoheadrightarrow \widehat{D}$ given by $h\mapsto h|D$ is a homeomorphism, and $G$ is determined if each dense subgroup of $G$ determines $G$. The principal result in this area, obtained independently by L. Außenhofer and M. J. Chasco, is the following: Every metrizable group is...

Several results are presented concerning the existence or nonexistence, for a subset S of ω₁, of a real r which works as a robust code for S with respect to a given sequence $⟨S_{\alpha }:\alpha <\omega ₁⟩$ of pairwise disjoint stationary subsets of ω₁, where “robustness” of r as a code may either mean that $S\in L[r,⟨S{*}_{\alpha }:\alpha <\omega ₁⟩]$ whenever each $S{*}_{\alpha }$ is equal to $S_{\alpha }$ modulo nonstationary changes, or may have the weaker meaning that $S\in L[r,⟨S_{\alpha }\cap C:\alpha <\omega ₁⟩]$ for every club C ⊆ ω₁. Variants of the above theme are also considered which result when the requirement that S gets exactly...

We define a new principle, SEP, which is true in all Cohen extensions of models of CH, and explore the relationship between SEP and other such principles. SEP is implied by each of CH*, the weak Freeze-Nation property of (ω), and the (ℵ₁,ℵ₀)-ideal property. SEP implies the principle $C{₂}^s\left(\omega ₂\right)$, but does not follow from $C{₂}^s\left(\omega ₂\right)$, or even $C^s\left(\omega ₂\right)$.

We examine the splitting number (B) and the reaping number (B) of quotient Boolean algebras B = (ω)/ℐ where ℐ is an $F_{\sigma }$ ideal or an analytic P-ideal. For instance we prove that under Martin’s Axiom ((ω)/ℐ) = for all $F_{\sigma }$ ideals ℐ and for all analytic P-ideals ℐ with the BW property (and one cannot drop the BW assumption). On the other hand under Martin’s Axiom ((ω)/ℐ) = for all $F_{\sigma }$ ideals and all analytic P-ideals ℐ (in this case we do not need the BW property). We also provide applications of these characteristics...