MAD FAMILIES, P +(I)-IDEALS AND IDEAL CONVERGENCE

Abstract

Let I be an ideal on ω, the notion of I-AD family was introducedin [3]. Analogous to the well studied ideal I(A) generated by almost disjointfamilies, we introduce and investigate the ideal I(I-A). It turns out that someproperties of I(I-A) depends on the structure of I. Denote a(I) the minimumof the cardinalities of infifinite I-MAD families, several characterizations fora(I) ≥ ω1 will be presented. Motivated by the work in [19], we introducethe cardinality sω,ω(I), and obtain a necessary condition for sω,ω(I) = s(I).As an application, we show fifinally that if a(I) ≥ s(I), then BW propertycoincides with Helly property.