A non-homogeneous zero-dimensional 𝑋 such that 𝑋×𝑋 is a group

Abstract

We provide an example of a zero-dimensional (separable metric) absolute Borel set X X which is not homogeneous, but whose square X × X X \times X admits the structure of a topological group. We also construct a zero-dimensional absolute Borel set Y Y such that Y Y is a homogeneous non-group but Y × Y Y \times Y is a group. This answers questions of Arhangel’skiĭ and Zhou.