More on Fréchet–urysohn Ideals

Abstract

We study the Rudin–Keisler pre-order on Fréchet–Urysohn ideals on $\omega $ . 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 $\mathcal {A},$ there is an AD family $\mathcal {B}$ such that $\mathcal {A}^{\perp } <_{{\textsf {RK}}}\mathcal {B}^{\perp }.$ • If $\mathcal {A}$ is a nowhere MAD family of size $\mathfrak {c}$ then there is a nowhere MAD family $\mathcal {B}$ such that $\mathcal {I}\left (\mathcal {A}\right ) $ and $\mathcal {I}\left ( \mathcal {B}\right ) $ are Rudin–Keisler incomparable. • There is a family $\left \{ \mathcal {B}_{\alpha }\mid \alpha \in \mathfrak {c}\right \} $ of nowhere MAD families such that if $\alpha \neq \beta $ , then $\mathcal {I}\left ( \mathcal {B}_{\alpha }\right ) $ and $\mathcal {I}\left ( \mathcal {B}_{\beta }\right ) $ are Rudin–Keisler incomparable. Here $\mathcal {I}(\mathcal {A})$ denotes the ideal generated by an AD family $\mathcal {A}$ . In the context of hyperspaces with the Vietoris topology, for a Fréchet–Urysohn-filter $\mathcal {F}$ we let $\mathcal {S}_{c}\left ( \mathcal {\xi }\left ( \mathcal {F}\right ) \right ) $ be the hyperspace of nontrivial convergent sequences of the space consisting of $\omega $ as discrete subset and only one accumulation point $\mathcal {F}$ whose neighborhoods are the elements of $\mathcal {F}$ together with the singleton $\{\mathcal {F}\}$ . For a FU-filter $\mathcal {F}$ we show that the following are equivalent: • $\mathcal {F}$ is a FUF-filter. • $\mathcal {S}_{c}\left ( \mathcal {\xi }\left ( \mathcal {F} \right ) \right ) $ is Baire.