We show that there is a $$ \varvec{\Sigma }_4^0$$ ideal such that it’s neither extendable to any $$ \varvec{\Pi }_3^0$$ ideal nor above the ideal $$ \textrm{Fin}\times \textrm{Fin} $$ in the sense of Katětov order, answering a question from M. Hrušák.

In this survey paper we collect several known results on destroying tall ideals on countable sets and maximal almost disjoint families with forcing. In most cases we provide streamlined proofs of the presented results. The paper contains results of many authors as well as a preview of results of a forthcoming paper of Brendle, Guzmán, Hrušák, and Raghavan.

We study the Mathias–Prikry and the Laver type forcings associated with filters and coideals. We isolate a crucial combinatorial property of Mathias reals, and prove that Mathias–Prikry forcings with summable ideals are all mutually bi-embeddable. We show that Mathias forcing associated with the complement of an analytic ideal always adds a dominating real. We also characterize filters for which the associated Mathias–Prikry forcing does not add eventually different reals, and show that they are countably generated provided they are Borel. We (...) give a characterization of ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\omega}$$\end{document}-hitting and ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\omega}$$\end{document}-splitting families which retain their property in the extension by a Laver type forcing associated with a coideal. (shrink)

We study the Katětov order on Borel ideals. We prove two structural theorems, one for Borel ideals, the other for analytic P-ideals. We isolate nine important Borel ideals and study the Katětov order among them. We also present a list of fundamental open problems concerning the Katětov order on Borel ideals.

We study some properties of the quotient forcing notions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${Q_{tr(I)} = \wp(2^{< \omega})/tr(i)}$$\end{document} and PI = B(2ω)/I in two special cases: when I is the σ-ideal of meager sets or the σ-ideal of null sets on 2ω. We show that the remainder forcing RI = Qtr(I)/PI is σ-closed in these cases. We also study the cardinal invariant of the continuum \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak{h}_{\mathbb{Q}}}$$\end{document}, the distributivity (...) number of the quotient \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${Dense(\mathbb{Q})/nwd}$$\end{document}, in order to show that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\wp(\mathbb{Q})/nwd}$$\end{document} collapses \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak{c}}$$\end{document} to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak{h}_{\mathbb{Q}}}$$\end{document}, thus answering a question addressed in Balcar et al. (Fundamenta Mathematicae 183:59–80, 2004). (shrink)

We investigate which filters onωcan contain towers, that is, a modulo finite descending sequence without any pseudointersection. We prove the following results:Many classical examples of nice tall filters contain no towers.It is consistent that tall analytic P-filters contain towers of arbitrary regular height.It is consistent that all towers generate nonmeager filters, in particular Borel filters do not contain towers.The statement “Every ultrafilter contains towers.” is independent of ZFC.Furthermore, we study many possible logical implications between the existence of towers in filters, (...) inequalities between cardinal invariants of filters $,${\rm{co}}{{\rm{f}}^{\rm{*}}}\left$,${\rm{no}}{{\rm{n}}^{\rm{*}}}\left$, and${\rm{co}}{{\rm{v}}^{\rm{*}}}\left$), and the existence of Luzin type families, that is, if${\cal F}$is a filter then${\cal X} \subseteq {[\omega ]^\omega }$is an${\cal F}$-Luzin family if$\left\{ {X \in {\cal X}:|X \setminus F| = \omega } \right\}$is countable for every$F \in {\cal F}$. (shrink)

The Steprāns forcing notion arises as quotient of the algebra of Borel sets modulo the ideal of σ-continuity of a certain Borel not σ-continuous function. We give a characterization of this forcing in the language of trees and use this characterization to establish such properties of the forcing as fusion and continuous reading of names. Although the latter property is usually implied by the fact that the associated ideal is generated by closed sets, we show that it is not the (...) case with Steprāns forcing. We also establish a connection between Steprāns forcing and Miller forcing thus giving a new description of the latter. Eventually, we exhibit a variety of forcing notions which do not have continuous reading of names in any presentation. (shrink)

It is relatively consistent with ZFC that every countable $FU_{fin} $ space of weight N₁ is metrizable. This provides a partial answer to a question of G. Gruenhage and P. Szeptycki [GS1].

We continue with the study of the Katětov order on MAD families. We prove that Katětov maximal MAD families exist under $\mathfrak {b=c}$ and that there are no Katětov-top MAD families assuming $\mathfrak {s\leq b}.$ This improves previously known results from the literature. We also answer a problem form Arciga, Hrušák, and Martínez regarding Katětov maximal MAD families.

We study the following natural strong variant of destroying Borel ideals: $\mathbb {P}$ $+$ -destroys $\mathcal {I}$ if $\mathbb {P}$ adds an $\mathcal {I}$ -positive set which has finite intersection with every $A\in \mathcal {I}\cap V$. Also, we discuss the associated variants $$ \begin{align*} \mathrm{non}^*(\mathcal{I},+)=&\min\big\{|\mathcal{Y}|:\mathcal{Y}\subseteq\mathcal{I}^+,\; \forall\;A\in\mathcal{I}\;\exists\;Y\in\mathcal{Y}\;|A\cap Y| \omega $ ; (4) we characterise when the Laver–Prikry, $\mathbb {L}(\mathcal {I}^*)$ -generic real $+$ -destroys $\mathcal {I}$, and in the case of P-ideals, when exactly $\mathbb {L}(\mathcal {I}^*)$ $+$ -destroys $\mathcal {I}$ ; (...) and (5) we briefly discuss an even stronger form of destroying ideals closely related to the additivity of the null ideal. (shrink)