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 $𝔯$.