Existence of an uncountable tower of Borel subgroups between the Prüfer group and the s-characterized group

Abstract

Recently in Dikranjan et al. (Fund Math 249: 185–209, 2020) an uncountable Borel subgroup $t^s_{(2^n)}({\mathbb T})$ (called statistically characterized subgroup) was constructed containing the Prüfer group ${\mathbb Z}(2^\infty )$ using the notion of statistical convergence. This note is based on the recent work (Bose et al. in Acta Math Hungar, 2020) which helps us to show that an uncountable chain of distinct Borel subgroups (each of size $\mathfrak {c}$) can be generated between ${\mathbb Z}(2^\infty )$ and $t^s_{(2^n)}({\mathbb T})$, whereas their intersection actually strictly contains the Prüfer group, with their union being strictly contained in $t^s_{(2^n)}({\mathbb T})$.