Special subsets of the generalized Cantor space $2^\kappa$ and generalized Baire space $\kappa^\kappa$

TL;DR

This paper explores generalized versions of classical special subsets in the Cantor and Baire spaces for uncountable cardinals, extending many known theorems and presenting open problems in this broader context.

Contribution

It provides a comprehensive catalogue of generalized classes of special subsets in $2^"kappa$ and $"kappa^"kappa$, including new results and open problems.

Findings

Many classical theorems extend to higher cardinals with additional assumptions.

The paper catalogs various generalized classes of special subsets.

Open problems are proposed for further research.

Abstract

In this paper, we are interested in parallels to the classical notions of special subsets in $\R$ defined in the generalized Cantor and Baire spaces ($2^\kappa$ and $\kappa^\kappa$). We consider generalizations of the well-known classes of special subsets, like Lusin sets, strongly null sets, concentrated sets, perfectly meagre sets, $\sigma$-sets, $\gamma$-sets, sets with Menger, Rothberger or Hurewicz property, but also of some less-know classes like $X$-small sets, meagre additive sets, Ramsey null sets, Marczewski, Silver, Miller and Laver-null sets. We notice that many classical theorems regarding these classes can be relatively easy generalized to higher cardinals although sometimes with some additional assumptions. This paper serves as a catalogue of such results along with some other generalizations and open problems.