On Borel ideals

Engelen, Fons van

doi:10.1016/0168-0072(94)90029-9

Cited by 19 publications

(25 citation statements)

References 14 publications

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“…is a group but X z X x X. By the results of [3], [XI = Da(Zj) for some 2 < a < w, and by [5] a is decomposable. Write X -Da(A,, y < a ) , and let D be the countable or a-compact subset of X given by Lemmas 3.1 and 3.2. a = n is finite and even.…”

Section: The Main Resultsmentioning

confidence: 99%

Borel Groups of Low Descriptive Complexity

Engelen

1996

Annals of the New York Academy of Sciences

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Section: The Main Resultsmentioning

confidence: 99%

Borel Groups of Low Descriptive Complexity

Engelen

1996

Annals of the New York Academy of Sciences

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“…About ideals on the ambiguous levels of the Borel hierarchy see [22]. We also present some (co)analytic examples.…”

Section: Some Non Tall Idealsmentioning

confidence: 99%

Towers in Filters, Cardinal Invariants, and Luzin Type Families

Brendle¹,

Farkas²,

Verner³

2018

J. symb. log.

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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 $F \in {\cal F}$.

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“…Thus, A ≈ A × A by Theorem 2.2, so A admits the structure of a topological group (in fact, of an ideal on ω) by [10]. We conclude that, in order to prove the theorem, it suffices to show that both Y × Y and X × X are everywhere properly D 2 (Σ 0 3 ), as they are clearly first category.…”

Section: The Examplesmentioning

confidence: 99%