## $\omega $-diagonalizability of $F_\sigma $ filters

### Volume 168 / 2022

Colloquium Mathematicum 168 (2022), 35-45
MSC: Primary 03E05; Secondary 03E15, 54H05, 91A44.
DOI: 10.4064/cm8416-2-2021
Published online: 10 August 2021

#### Abstract

We prove, without any form of determinacy, that $F_\sigma $ filters are exactly filters $\omega $-diagonalizable by $\mathcal F ^+$-universal sets. For analytic filters, $F_\sigma $ filters are exactly $P^+$(tree)-filters (this is the extension to analytic ideals of a theorem about Borel ideals by Hrušák and Meza-Alcántara). We also show some conditions equivalent to the fact that a filter is a subset of a proper $F_\sigma $ filter.