We generalize the Bolzano-Weierstrass theorem on ideal convergence. We show examples of ideals with and without the Bolzano-Weierstrass property, and give characterizations of BW property in terms of submeasures and extendability to a maximal P-ideal. We show applications to Rudin-Keisler and Rudin-Blass orderings of ideals and quotient Boolean algebras. In particular we show that an ideal does not have BW property if and only if its quotient Boolean algebra has a countably splitting family.

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)

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 ultrafilters onω2produced by forcing with the quotient of${\cal P}$(ω2) by the Fubini square of the Fréchet filter onω. We show that such an ultrafilter is a weak P-point but not a P-point and that the only nonprincipal ultrafilters strictly below it in the Rudin–Keisler order are a single isomorphism class of selective ultrafilters. We further show that it enjoys the strongest square-bracket partition relations that are possible for a non-P-point. We show that it is not basically generated but (...) that it shares with basically generated ultrafilters the property of not being at the top of the Tukey ordering. In fact, it is not Tukey-above [ω1]<ω, and it has only continuum many ultrafilters Tukey-below it. A tool in our proofs is the analysis of similar (but not the same) properties for ultrafilters obtained as the sum, over a selective ultrafilter, of nonisomorphic selective ultrafilters. (shrink)

We study the $I$-ultrafilters on $\omega$, where $I$ is a collection of subsets of a set $X$, usually $\mathbb{R}$ or $\omega_1$. The $I$-ultrafilters usually contain the $P$-points, often as a small proper subset. We study relations between $I$-ultrafilters for various $I$, and closure of $I$-ultrafilters under ultrafilter sums. We consider, but do not settle, the question whether $I$-ultrafilters always exist.

We study the I-ultrafilters on ω, where I is a collection of subsets of a set X, usually R or ω 1 . The I-ultrafilters usually contain the P-points, often as a small proper subset. We study relations between I-ultrafilters for various I, and closure of I-ultrafilters under ultrafilter sums. We consider, but do not settle, the question whether I-ultrafilters always exist.