Small Hurewicz and Menger sets which have large continuous images

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.

💡 Research Summary

The paper investigates the interplay between classical selection principles—Hurewicz, Menger, and the stronger property (S_{1}(\Gamma,\mathcal O))—and the notion of a set of reals that contains no perfect subset (i.e., a totally imperfect set). The authors develop new construction techniques that, under certain cardinal‑invariant assumptions, produce totally imperfect subsets of the Cantor space (2^{\omega}) (or of its square) which are Hurewicz or Menger, yet admit a continuous surjection onto the whole Cantor space. This is striking because previously known constructions of totally imperfect Hurewicz or Menger sets always satisfied (S_{1}(\Gamma,\mathcal O)) and could not be mapped onto a perfect set.

The main results are:

1. Theorem 2.1 (Hurewicz case). Assuming (\operatorname{cov}(\mathcal N)=\mathfrak b=\mathfrak c) (which holds under CH), the authors construct a set (X\subseteq2^{\omega}\times2^{\omega}) such that:
   • (X) is Hurewicz,
   • (X) is a (\lambda’)-set (every countable subset is a (G_{\delta}) in the relative topology),
   • (X) is totally imperfect,
   • the projection (\pi