The Baire property on the hyperspace of nontrivial convergent sequences

Garcı́a-Ferreira, S.; Rojas-Hernández, R.; Ortiz-Castillo, Y. F.

doi:10.1016/j.topol.2020.107505

Cited by 2 publications

(7 citation statements)

References 15 publications

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“…Let X be a Fréchet–Urysohn space that has a dense set D of isolated points. The topological game introduced in [7] is defined as follows:…”

Section: The Hyperspace Of Convergent Sequencesmentioning

confidence: 99%

“…The systematic study of the hyperspace of nontrivial convergent sequences of a Fréchet–Urysohn nondiscrete space X was initiated in [4], where is equipped with the Vietoris topology. The categorical properties on , together with other topological properties, were considered in [7]. Indeed, in that paper, it was proved that is never a Baire space when the space X is crowded, and this result was improved in [7] by showing that is meager whenever X is crowded (this assertion was also proved independently in [18]).…”

Section: Introductionmentioning

confidence: 99%

“…The categorical properties on , together with other topological properties, were considered in [7]. Indeed, in that paper, it was proved that is never a Baire space when the space X is crowded, and this result was improved in [7] by showing that is meager whenever X is crowded (this assertion was also proved independently in [18]). Hence, if is Baire, then X has a dense subset of isolated points.…”

Section: Introductionmentioning

confidence: 99%

“…Hence, if is Baire, then X has a dense subset of isolated points. Nontrivial examples of spaces X for which is Baire were given in [7]. The authors of [7, Proposition 2.8] proposed the following problem:…”

Section: Introductionmentioning

confidence: 99%

“…Nontrivial examples of spaces X for which S c (X ) is Baire were given in [7]. The authors of [7,Proposition 2.8] proposed the following problem: Problem 1.2. Determine the FU-filters F on for which the space S c ( (F)) is Baire.…”

mentioning

confidence: 99%

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More on Fréchet–urysohn Ideals

Ferreira¹,

Guzmán²

2021

J. symb. log.

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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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“…Let X be a Fréchet–Urysohn space that has a dense set D of isolated points. The topological game introduced in [7] is defined as follows:…”

Section: The Hyperspace Of Convergent Sequencesmentioning

confidence: 99%