Recall that a P-set is a closed set X such that the intersection of countably many neighborhoods of X is again a neighborhood of X. We show that if 𝔱 = 𝔠 then there is a minimal right ideal of (βℕ,+) that is also a P-set. We also show that the existence of such P-sets implies the existence of P-points; in particular, it is consistent with ZFC that no minimal right ideal is a P-set. As an application of these results, we prove that it is both consistent with and independent of ZFC that the shift...

We investigate families of partitions of ω which are related to special coideals, so-called happy families, and give a dual form of Ramsey ultrafilters in terms of partitions. The combinatorial properties of these partition-ultrafilters, which we call Ramseyan ultrafilters, are similar to those of Ramsey ultrafilters. For example it will be shown that dual Mathias forcing restricted to a Ramseyan ultrafilter has the same features as Mathias forcing restricted to a Ramsey ultrafilter. Further we...

Some constructions of spaces with/without dense subspaces satisfying stronger separation axioms are presented.

The Rothberger number (ℐ) of a definable ideal ℐ on ω is the least cardinal κ such that there exists a Rothberger gap of type (ω,κ) in the quotient algebra (ω)/ℐ. We investigate (ℐ) for a class of $F_{\sigma }$ ideals, the fragmented ideals, and prove that for some of these ideals, like the linear growth ideal, the Rothberger number is ℵ₁, while for others, like the polynomial growth ideal, it is above the additivity of measure. We also show that it is consistent that there are infinitely many (even continuum...

Given a countable Borel equivalence relation, I introduce an invariant measuring how difficult it is to find Borel sets separating its equivalence classes. I evaluate these invariants in several standard generic extensions.

The general question of when a countably compact topological space is sequentially compact, or has a nontrivial convergent sequence, is studied from the viewpoint of basic cardinal invariants and small uncountable cardinals. It is shown that the small uncountable cardinal 𝔥 is both the least cardinality and the least net weight of a countably compact space that is not sequentially compact, and that it is also the least hereditary Lindelöf degree in most published models. Similar results, some definitive,...

We give an equivalent, but simpler formulation of the axiom SEP, which was introduced in [9] in order to capture some of the combinatorial behaviour of models of set theory obtained by adding Cohen reals to a model of CH. Our formulation shows that many of the consequences of the weak Freese-Nation property of 𝒫(ω) studied in [6] already follow from SEP. We show that it is consistent that SEP holds while 𝒫(ω) fails to have the (ℵ₁,ℵ ₀)-ideal property introduced in [2]. This answers a question...

In the present paper we introduce a convergence condition $\left({\Sigma }^{\text{'}}\right)$ and continue the study of “not distinguish” for various kinds of convergence of sequences of real functions on a topological space started in [2] and [3]. We compute cardinal invariants associated with introduced properties of spaces.

Jörg Brendle (2003) used Hechler’s forcing notion for adding a maximal almost disjoint family along an appropriate template forcing construction to show that (the minimal size of a maximal almost disjoint family) can be of countable cofinality. The main result of the present paper is that ${}_g$, the minimal size of a maximal cofinitary group, can be of countable cofinality. To prove this we define a natural poset for adding a maximal cofinitary group of a given cardinality, which enjoys certain combinatorial...

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

We show that cov(M) is the least infinite cardinal λ such that $P_{\omega }\left(\lambda \right)$ (the set of all finite subsets of λ ) fails to satisfy a certain natural generalization of Ramsey’s Theorem.

The linear refinement number is the minimal cardinality of a centered family in ${\left[\omega \right]}^{\omega }$ such that no linearly ordered set in $({\left[\omega \right]}^{\omega },\subseteq *)$ refines this family. The linear excluded middle number is a variation of . We show that these numbers estimate the critical cardinalities of a number of selective covering properties. We compare these numbers to the classical combinatorial cardinal characteristics of the continuum. We prove that = = in all models where the continuum is at most ℵ₂, and that the cofinality of is...

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...

We show the consistency of the statement: "the set of regular cardinals which are the characters of ultrafilters on ω is not convex". We also deal with the set of π-characters of ultrafilters on ω.

Let χ be the minimum cardinality of a subset of ${}^{\omega }2$ that cannot be made convergent by multiplication with a single Toeplitz matrix. By an application of a creature forcing we show that < χ is consistent. We thus answer a question by Vojtáš. We give two kinds of models for the strict inequality. The first is the combination of an ℵ₂-iteration of some proper forcing with adding ℵ₁ random reals. The second kind of models is obtained by adding δ random reals to a model of $M{A}_{<\kappa }$ for some δ ∈ [ℵ₁,κ). It...

The $\sigma$-property of a Riesz space (real vector lattice) $B$ is: For each sequence $\left\{b_n\right\}$ of positive elements of $B$, there is a sequence $\left\{{\lambda }_n\right\}$ of positive reals, and $b\in B$, with ${\lambda }_n{b}_n\le b$ for each $n$. This condition is involved in studies in Riesz spaces of abstract Egoroff-type theorems, and of the countable lifting property. Here, we examine when “$\sigma$” obtains for a Riesz space of continuous real-valued functions $C\left(X\right)$. A basic result is: For discrete $X$, $C\left(X\right)$ has $\sigma$ iff the cardinal $\left|X\right|<𝔟$, Rothberger’s bounding number. Consequences and...