Subgyrogroups within the product spaces of paratopological gyrogroups

We present a characterization of paratopological gyrogroups that can be topologically embedded as subgyrogroups into a product of first-countable Tᵢ 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 T₁ strongly paratopological gyrogroups if and only if G is T₁, $ω$-balanced and the weakly Hausdorff number of G is countable. This means that for every neighborhood U of the identity 0 in G, there exists a countable family $γ$ of neighborhoods of 0 such that for all V ∈γ, V∈γ (⊖ V)⊆ U. 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. This means that for every neighborhood U of the identity 0 in G, there exists a countable family $γ$ of neighborhoods of 0 such that for all V ∈γ, V∈γ (V V)⊆ U.