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Small Hurewicz and Menger sets which have large continuous images

Piotr Szewczak, Tomasz Weiss, Lyubomyr Zdomskyy

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 $\mathsf{S}_1(Γ,\mathrm{O})$ 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.

Small Hurewicz and Menger sets which have large continuous images

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 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.
Paper Structure (11 sections, 17 theorems, 29 equations, 2 figures)

This paper contains 11 sections, 17 theorems, 29 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Hurewicz sets
  3. Menger sets
  4. More consequences
  5. On the problem of Nowik and Tsaban
  6. Everywhere meager and everywhere null
  7. Comments and open problems
  8. Groupability and considered properties
  9. The property $\mathsf{U}_\mathrm{id}(\Gamma,\Gamma)$
  10. Separation of properties
  11. Around PMT sets

Key Result

Theorem 1.1

Each Hurewicz set in ${2^\omega}$ which cannot be mapped continuously onto ${2^\omega}$ is PMT.

Figures (2)

  • Figure :
  • Figure :

Theorems & Definitions (31)

  • Theorem 1.1: Nowik nowik
  • Theorem 2.1
  • Remark 2.2
  • Lemma 2.3: NW
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 21 more