Concentrated sets and γ-sets in the Miller model

Valentin Haberl; Piotr Szewczak; Lyubomyr Zdomskyy

doi:10.1016/j.topol.2025.109503

Concentrated sets and γ-sets in the Miller model

Valentin Haberl
, Piotr Szewczak
, Lyubomyr Zdomskyy

Research output: Contribution to journal › Article › peer-review

Abstract

Using combinatorial covering properties, we show that there is no concentrated set of reals of size ω₂ in the Miller model. The main result refutes a conjecture of Bartoszyński and Halbeisen. We also prove that there are no γ -set of reals of size ω₂ in the Miller model.

Original language	English
Article number	109503
Journal	Topology and its Applications
Volume	379
DOIs	https://doi.org/10.1016/j.topol.2025.109503
State	Published - 15 Feb 2026
Externally published	Yes

Bibliographical note

Publisher Copyright:
© 2025 The Author(s).

Keywords

Concentrated set
Hurewicz space
Miller forcing
Rothberger space

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10.1016/j.topol.2025.109503License: CC BY

Cite this

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title = "Concentrated sets and γ-sets in the Miller model",

abstract = "Using combinatorial covering properties, we show that there is no concentrated set of reals of size ω2 in the Miller model. The main result refutes a conjecture of Bartoszy{\'n}ski and Halbeisen. We also prove that there are no γ -set of reals of size ω2 in the Miller model.",

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AU - Haberl, Valentin

AU - Szewczak, Piotr

AU - Zdomskyy, Lyubomyr

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N2 - Using combinatorial covering properties, we show that there is no concentrated set of reals of size ω2 in the Miller model. The main result refutes a conjecture of Bartoszyński and Halbeisen. We also prove that there are no γ -set of reals of size ω2 in the Miller model.

AB - Using combinatorial covering properties, we show that there is no concentrated set of reals of size ω2 in the Miller model. The main result refutes a conjecture of Bartoszyński and Halbeisen. We also prove that there are no γ -set of reals of size ω2 in the Miller model.

KW - Concentrated set

KW - Hurewicz space

KW - Miller forcing

KW - Rothberger space

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Concentrated sets and γ-sets in the Miller model

Abstract

Bibliographical note

Keywords

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