Strong meager properties for filters

Abstract

We analyze several ``strong meager'' properties for filters on the natural numbers between the classical Baire property and a filter being $F_{\sigma }$. Two such properties have been studied by Talagrand and a few more combinatorial ones are investigated. In particular, we define the notion of a P${}^{+}$-filter, a generalization of the traditional concept of P-filter, and prove the existence of a non-meager P${}^{+}$-filter. Our motivation lies in understanding the structure of filters generated by complements of members of a maximal almost disjoint family.