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MORE ON FRÉCHET–URYSOHN IDEALS
Published online by Cambridge University Press: 10 June 2021
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.
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- Copyright
- © The Author(s), 2021. Published by Cambridge University Press on behalf of The Association for Symbolic Logic
References
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