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

Abstract

In this paper, we are interested in parallels to the classical notions of special subsets in $\mathbb{R}$ 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.