Towers in Filters, Cardinal Invariants, and Luzin Type Families

Abstract

We investigate which filters on ω can contain towers, that is, a modulo finite descending sequence without any pseudointersection (in ${[\omega ]^\omega }$). We prove the following results:(1)Many classical examples of nice tall filters contain no towers (in ZFC).(2)It is consistent that tall analytic P-filters contain towers of arbitrary regular height (simultaneously for many regular cardinals as well).(3)It is consistent that all towers generate nonmeager filters (this answers a question of P. Borodulin-Nadzieja and D. Chodounský), in particular (consistently) Borel filters do not contain towers.(4)The statement “Every ultrafilter contains towers.” is independent of ZFC (this improves an older result of K. Kunen, J. van Mill, and C. F. Mills).Furthermore, we study many possible logical (non)implications between the existence of towers in filters, inequalities between cardinal invariants of filters (${\rm{ad}}{{\rm{d}}^{\rm{*}}}\left( {\cal F} \right)$, ${\rm{co}}{{\rm{f}}^{\rm{*}}}\left( {\cal F} \right)$, ${\rm{no}}{{\rm{n}}^{\rm{*}}}\left( {\cal F} \right)$, and ${\rm{co}}{{\rm{v}}^{\rm{*}}}\left( {\cal F} \right)$), and the existence of Luzin type families (of size $\ge {\omega _2}$), 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 $F \in {\cal F}$.

Ideal approach to convergence in functional spaces

“…Thus we distinguish an [Ω, I-Γ]-space for many of the pairs of ideals among standard critical ideals considered in [6,7,19,20], see Corollaries 6.5, 8.4, and Diagram 3. Moreover, we study similar variation of Fréchet-Urysohn property of C p (X), denoted [Ω 0 , I-Γ 0 ]-space, and introduced by P. Borodulin-Nadzieja and B. Farkas [5].…”

“…In the first part of the section, we recall a basic terminology on ideals on natural numbers, which mainly follows [6,7,19,20]. Afterwards, in the rest of the section, Katětov power ideals are introduced and discussed in more detail.…”

“…Note that if A 1 ⊇ Fin and A 1 ≤ ϕ A 2 , then ϕ is A 2 -to-one. In Diagram 2, we summarize relations between standard Borel ideals on ω studied in [6,7,19,20] adding Katětov power ideals which play an important role in the study of δ-sets, see below. Here an arrow means ≤ K .…”

“…M and Fin α N are (1-1)-equivalent. 7 Although m does not carry any ideal according to our definition, ∅m is hereditary, closed under finite unions, and m / ∈ ∅m. Moreover, our definition of…”

“…It is not known whether Ran ≤ K S, see[6,7,19,20] 18. Under assumption that we added all arrows arising from transitivity.…”