We compare the forcing-related properties of a complete Boolean algebra $𝔹$ with the properties of the convergences ${\lambda }_{\mathrm{s}}$ (the algebraic convergence) and ${\lambda }_{\mathrm{ls}}$ on $𝔹$ generalizing the convergence on the Cantor and Aleksandrov cube, respectively. In particular, we show that ${\lambda }_{\mathrm{ls}}$ is a topological convergence iff forcing by $𝔹$ does not produce new reals and that ${\lambda }_{\mathrm{ls}}$ is weakly topological if $𝔹$ satisfies condition $\left(\hslash \right)$ (implied by the $𝔱$-cc). On the other hand, if ${\lambda }_{\mathrm{ls}}$ is a weakly topological convergence, then $𝔹$ is a $2^{𝔥}$-cc algebra...

We prove among other theorems that it is consistent with $ZFC$ that there exists a set $X\subseteq 2^{\omega }$ which is not meager additive, yet it satisfies the following property: for each $F_{\sigma }$ measure zero set $F$, $X+F$ belongs to the intersection ideal $ℳ\cap 𝒩$.

In this paper we develop the semifilter approach to the classical Menger and Hurewicz properties and show that the small cardinal $𝔤$ is a lower bound of the additivity number of the $\sigma$-ideal generated by Menger subspaces of the Baire space, and under $𝔲<𝔤$ every subset $X$ of the real line with the property $Split(\Lambda ,\Lambda )$ is Hurewicz, and thus it is consistent with ZFC that the property $Split(\Lambda ,\Lambda )$ is preserved by unions of less than $𝔟$ subsets of the real line.

Developing the idea of assigning to a large cover of a topological space a corresponding semifilter, we show that every Menger topological space has the property ${\bigcup }_{fin}(𝒪,T^{*})$ provided $(𝔲<𝔤)$, and every space with the property ${\bigcup }_{fin}(𝒪,T^{*})$ is Hurewicz provided $({Depth}^{+}\left({\left[\omega \right]}^{{\aleph }_0}\right)\le 𝔟)$. Combining this with the results proven in cited literature, we settle all questions whether (it is consistent that) the properties $\text{P}$ and $\text{Q}$ [do not] coincide, where $\text{P}$ and $\text{Q}$ run over ${\bigcup }_{fin}(𝒪,\Gamma )$, ${\bigcup }_{fin}(𝒪,T)$, ${\bigcup }_{fin}(𝒪,T^{*})$, ${\bigcup }_{fin}(𝒪,\Omega )$, and ${\bigcup }_{fin}(𝒪,𝒪)$.

We prove an analogue to Dordal’s result in P.L. Dordal, A model in which the base-matrix tree cannot have cofinal branches, J. Symbolic Logic 52 (1980), 651–664. He obtained a model of ZFC in which there is a tree $\pi$-base for ${\mathbb{N}}^{*}$ with no ${\omega }_2$ branches yet of height ${\omega }_2$. We establish that this is also possible for ${ℝ}^{*}$ using a natural modification of Mathias forcing.

We define two cardinal invariants of the continuum which arise naturally from combinatorially and topologically appealing properties of almost disjoint families of sets of the natural numbers. These are the never soft and never countably paracompact numbers. We show that these cardinals must both be equal to ${\omega }_1$ under the effective weak diamond principle $♦(\omega ,\omega ,<)$, answering questions of da Silva S.G., On the presence of countable paracompactness, normality and property $\left(a\right)$ in spaces from almost disjoint families,...

A new ⋄-like principle ${\diamond }_{}$ consistent with the negation of the Continuum Hypothesis is introduced and studied. It is shown that $\neg {\diamond }_{}$ is consistent with CH and that in many models of = ω₁ the principle ${\diamond }_{}$ holds. As ${\diamond }_{}$ implies that there is a MAD family of size ℵ₁ this provides a partial answer to a question of J. Roitman who asked whether = ω₁ implies = ω₁. It is proved that ${\diamond }_{}$ holds in any model obtained by adding a single Laver real, answering a question of J. Brendle who asked whether = ω₁ in such models....

It is consistent that $${\aleph }_1<\mathrm{add}\left(𝒩\right)<\mathrm{add}\left(ℳ\right)=𝔟<\mathrm{cov}\left(𝒩\right)<\mathrm{non}\left(ℳ\right)<\mathrm{cov}\left(ℳ\right)=2^{{\aleph }_0}.$$ Assuming four strongly compact cardinals, it is consistent that $$\begin{array}{ccc}\hfill {\aleph }_1& <\mathrm{add}\left(𝒩\right)<\mathrm{add}\left(ℳ\right)=𝔟<\mathrm{cov}\left(𝒩\right)<\mathrm{non}\left(ℳ\right)\hfill & \hfill <\mathrm{cov}\left(ℳ\right)<\mathrm{non}\left(𝒩\right)<\mathrm{cof}\left(ℳ\right)=𝔡<\mathrm{cof}\left(𝒩\right)<2^{{\aleph }_0}.\end{array}$$

We show the consistency of "there is a nice σ-ideal ℐ on the reals with add(ℐ) = ℵ₁ which cannot be represented as the union of a strictly increasing sequence of length ω₁ of σ-subideals". This answers [Borodulin-Nadzieja and Głąb, Math. Logic Quart. 57 (2011), 582-590, Problem 6.2(ii)].

We prove several results on some cardinal invariants of the continuum which are closely related to either the splitting number $𝔰$ or its dual, the reaping number $𝔯$.

We discuss the Borel Tukey ordering on cardinal invariants of the continuum. We observe that this ordering makes sense for a larger class of cardinals than has previously been considered. We then provide a Borel version of a large portion of van Douwen's diagram. For instance, although the usual proof of the inequality 𝔭 ≤ 𝔟 does not provide a Borel Tukey map, we show that in fact there is one. Afterwards, we revisit a result of Mildenberger concerning a generalization of the unsplitting and splitting...

A hull of A ⊆ [0,1] is a set H containing A such that λ*(H) = λ*(A). We investigate all four versions of the following problem. Does there exist a monotone (with respect to inclusion) map that assigns a Borel/$G_{\delta }$ hull to every negligible/measurable subset of [0,1]? Three versions turn out to be independent of ZFC, while in the fourth case we only prove that the nonexistence of a monotone $G_{\delta }$ hull operation for all measurable sets is consistent. It remains open whether existence here is also consistent....

We calculate the cardinal characteristics of the $\sigma$-ideal $\mathscr{H}𝒩\left(G\right)$ of Haar null subsets of a Polish non-locally compact group $G$ with invariant metric and show that $cov\left(\mathscr{H}𝒩\right(G\left)\right)\le 𝔟\le max\{𝔡,non(𝒩\left)\right\}\le non\left(\mathscr{H}𝒩\right(G\left)\right)\le cof\left(\mathscr{H}𝒩\right(G\left)\right)>min\{𝔡,non(𝒩\left)\right\}$. If $G={\prod }_{n\ge 0}{G}_n$ is the product of abelian locally compact groups $G_n$, then $add\left(\mathscr{H}𝒩\right(G\left)\right)=add\left(𝒩\right)$, $cov\left(\mathscr{H}𝒩\right(G\left)\right)=min\{𝔟,cov(𝒩\left)\right\}$, $non\left(\mathscr{H}𝒩\right(G\left)\right)=max\{𝔡,non(𝒩\left)\right\}$ and $cof\left(\mathscr{H}𝒩\right(G\left)\right)\ge cof\left(𝒩\right)$, where $𝒩$ is the ideal of Lebesgue null subsets on the real line. Martin Axiom implies that $cof\left(\mathscr{H}𝒩\left(G\right)\right)>2^{{\aleph }_0}$ and hence $G$ contains a Haar null subset that cannot be enlarged to a Borel or projective Haar null subset of $G$. This gives a negative (consistent) answer to a question of...

A graph G on a topological space X as its set of vertices is clopen if the edge relation of G is a clopen subset of X² without the diagonal. We study clopen graphs on Polish spaces in terms of their finite induced subgraphs and obtain information about their cochromatic numbers. In this context we investigate modular profinite graphs, a class of graphs obtained from finite graphs by taking inverse limits. This continues the investigation of continuous colorings on Polish spaces and their homogeneity...

We investigate the structure of the Tukey ordering among directed orders arising naturally in topology and measure theory.

We study combinatorial principles we call the Homogeneity Principle HP(κ) and the Injectivity Principle IP(κ,λ) for regular κ > ℵ₁ and λ ≤ κ which are formulated in terms of coloring the ordinals < κ by reals. These principles are strengthenings of $C^s\left(\kappa \right)$ and $F^s\left(\kappa \right)$ of I. Juhász, L. Soukup and Z. Szentmiklóssy. Generalizing their results, we show e.g. that IP(ℵ₂,ℵ₁) (hence also IP(ℵ₂,ℵ₂) as well as HP(ℵ₂)) holds in a generic extension of a model of CH by Cohen forcing, and IP(ℵ₂,ℵ₂) (hence also HP(ℵ₂))...

We study combinatorial properties of the partial order (Dense(ℚ),⊆). To do that we introduce cardinal invariants ${}_{\mathbb{Q}}$, ${}_{\mathbb{Q}}$, ${}_{\mathbb{Q}}$, ${}_{\mathbb{Q}}$, ${}_{\mathbb{Q}}$, ${}_{\mathbb{Q}}$ describing properties of Dense(ℚ). These invariants satisfy ${}_{\mathbb{Q}}$ ≤ ℚ ≤ ℚ ≤ ℚ ≤ ℚ ≤ ℚ$.WecomparethemwiththeiranaloguesinthewellstudiedBooleanalgebra\left(\omega \right)/fin.Weshowthat$ℚ = p$,$ℚ = t$and$ℚ = i$,whereas$ℚ > h$and$ℚ > r$arebothshowntoberelativelyconsistentwithZFC.Wealsoinvestigatecombinatoricsoftheidealnwdofnowheredensesubsetsof,\mathbb{Q}.Inparticular,weshowthat$non(M)=min||: ⊆ Dense(R) ∧ (∀I ∈ nwd(R))(∃D ∈ )(I ∩ D = ∅) and cof(M) = min||: ⊆ Dense(ℚ) ∧ (∀I ∈ nwd)(∃D ∈ )(I ∩ = ∅). We use these facts to show that cof(M) ≤ i, which improves a result of S. Shelah.