Small Hurewicz and Menger sets which have large continuous images

Piotr Szewczak; Tomasz Weiss; Lyubomyr Zdomskyy

doi:10.1016/j.apal.2026.103746

Small Hurewicz and Menger sets which have large continuous images

Piotr Szewczak
, Tomasz Weiss
, Lyubomyr Zdomskyy

University of Warsaw
Cardinal Stefan Wyszynski University
TU Wien

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

Abstract

We provide new techniques to construct sets of reals without perfect subsets and with the Hurewicz or Menger covering properties. In particular, we show that if the Continuum Hypothesis holds, then there are such sets which can be mapped continuously onto the Cantor space. These results allow to separate the properties of Menger and S₁(Γ,O) in the realm of sets of reals without perfect subsets and solve a problem of Nowik and Tsaban concerning perfectly meager subsets in the transitive sense. We present also some other applications of the mentioned above methods.

Original language	English
Article number	103746
Journal	Annals of Pure and Applied Logic
Volume	177
Issue number	8
DOIs	https://doi.org/10.1016/j.apal.2026.103746
State	Published - 1 Aug 2026
Externally published	Yes

Bibliographical note

Publisher Copyright:
© 2026 Elsevier B.V.

Keywords

Countably perfectly meager
Everywhere meager
Hurewicz
Menger
Perfectly meager
Totally imperfect

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10.1016/j.apal.2026.103746

Cite this

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AU - Weiss, Tomasz

AU - Zdomskyy, Lyubomyr

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AB - We provide new techniques to construct sets of reals without perfect subsets and with the Hurewicz or Menger covering properties. In particular, we show that if the Continuum Hypothesis holds, then there are such sets which can be mapped continuously onto the Cantor space. These results allow to separate the properties of Menger and S1(Γ,O) in the realm of sets of reals without perfect subsets and solve a problem of Nowik and Tsaban concerning perfectly meager subsets in the transitive sense. We present also some other applications of the mentioned above methods.

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KW - Everywhere meager

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KW - Menger

KW - Perfectly meager

KW - Totally imperfect

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ER -

Small Hurewicz and Menger sets which have large continuous images

Abstract

Bibliographical note

Keywords

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