We investigate the Tukey order in the class of Fσ ideals of subsets of ω. We show that no nontrivial Fσ ideal is Tukey below a Gδ ideal of compact sets. We introduce the notions of flat ideals and gradually flat ideals. We prove a dichotomy theorem for flat ideals isolating gradual flatness as the side of the dichotomy that is structurally good. We give diverse characterizations of gradual flatness among flat ideals using Tukey reductions and games. For example, we (...) show that gradually flat ideals are precisely those flat ideals that are Tukey below the ideal of density zero sets. (shrink)

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 paper, we construct group trees of tame Polish groups of the form, where are countable abelian. Our first example where all are torsion reaches the known upper bound. Next we give examples having height for any, close to the known bound.

PurposeOlder adults aged 60 years and above are classified as being of high-risk for infection during the COVID-19 pandemic. This study aimed to investigate the associations of psychological factors of preventive behaviors with three preventive behaviors among Chinese older adults during the COVID-19 pandemic.MethodsA cross-sectional questionnaire survey was administered via SOJUMP, a widely used online survey platform in China. A total of 928 older adults were recruited using a snowball sampling approach from Hubei Province and outside Hubei Province in China (...) during May 18, 2020 to June 7, 2020. Multiple hierarchical regressions were conducted with four models to examine the association between demographic, past behavior, psychological factors and each preventive behavior.Principal FindingsAll three preventive behaviors in older adults increased dramatically during the pandemic of COVID-19. Gender, living status, educational level, past behavior, health knowledge, intention and planning significantly predicted hand washing behavior, R2 = 0.395, F = 54.372, p < 0.001. Gender, education level, important others infection, past behavior, health knowledge, planning and action control significantly predicted mask wearing behavior, R2 = 0.202, F = 23.197, p < 0.001. Living place, past behavior and health knowledge significantly predicted social distancing behavior, R2 = 0.204, F = 26.201, p < 0.001.Major ConclusionsPast behavior and health knowledge predicted all three preventive behaviors. Planning was an important psychological factor for both hand washing and mask wearing behaviors. All those critical demographic and psychological factors are critical for future interventions to facilitate older adults to comply with three preventive behaviors in daily life and to stay healthy during the COVID-19 pandemic. (shrink)

A P-point ultrafilter over \(\omega \) is called an interval P-point if for every function from \(\omega \) to \(\omega \) there exists a set _A_ in this ultrafilter such that the restriction of the function to _A_ is either a constant function or an interval-to-one function. In this paper we prove the following results. (1) Interval P-points are not isomorphism invariant under \(\textsf{CH}\) or \(\textsf{MA}\). (2) We identify a cardinal invariant \(\textbf{non}^{**}({\mathcal {I}}_{\tiny {\hbox {int}}})\) such that every filter base (...) of size less than continuum can be extended to an interval P-point if and only if \(\textbf{non}^{**}({\mathcal {I}}_{\tiny {\hbox {int}}})={\mathfrak {c}}\). (3) We prove the generic existence of slow/rapid non-interval P-points and slow/rapid interval P-points which are neither quasi-selective nor weakly Ramsey under the assumption \({\mathfrak {d}}={\mathfrak {c}}\) or \(\textbf{cov}({\mathcal {B}})={\mathfrak {c}}\). (shrink)