Type: Article

More on Fréchet-Urysohn ideals

García-Ferreira, Salvador; Guzmán, Osvaldo;

Abstract:

We study the Rudin–Keisler pre-order on Fréchet–Urysohn ideals on ? . We solve three open questions posed by S. García-Ferreira and J. E. Rivera-Gómez in the articles [5] and [6] by establishing the following results: • For every AD family A, there is an AD family B such that A?<RKB?. • If A is a nowhere MAD family of size c then there is a nowhere MAD family B such that I(A) and I(B) are Rudin–Keisler incomparable. • There is a family {B????c} of nowhere MAD families such that if ??? , then I(B?) and I(B?) are Rudin–Keisler incomparable. Here I(A) denotes the ideal generated by an AD family A . In the context of hyperspaces with the Vietoris topology, for a Fréchet–Urysohn-filter F we let Sc(?(F)) be the hyperspace of nontrivial convergent sequences of the space consisting of ? as discrete subset and only one accumulation point F whose neighborhoods are the elements of F together with the singleton {F} . For a FU-filter F we show that the following are equivalent: • F is a FUF-filter. • Sc(?(F)) is Baire.