Subgyrogroups within the product spaces of paratopological gyrogroups

Abstract

We present a characterization of paratopological gyrogroups that can be topologically embedded as subgyrogroups into a product of first-countable T_i paratopological gyrogroups for i = 0, 1, 2. Specifically, we demonstrate that a strongly paratopological gyrogroup G is topologically isomorphic to a subgyrogroup of a topological product of first-countable T ₁ strongly paratopological gyrogroups if and only if G is T ₁, ω-balanced and the weakly Hausdorff number of G is countable. Similarly, we prove that a strongly paratopological gyrogroup G is topologically isomorphic to a subgyrogroup of a topological product of first-countable Hausdorff strongly paratopological gyrogroups if and only if G is Hausdorff, ω-balanced and the Hausdorff number of G is countable.