Uniformities on strongly topological gyrogroups

Abstract In this paper, we introduced three uniformities V G l , V G r and V G which are induced in a natural way on a strongly topological gyrogroup ( G , ⊕ , τ ) . It is mainly proved that (1) each of the three uniformities is compatible with G; (2) if H is a subgyrogroup of G, then V G , H l = V H l , V G , H r = V H r and V G , H = V H ; (3) if { ( G i , ⊕ i , τ i ) : i ∈ I } is a family of strongly topological gyrogroups, then V G l = ∏ i ∈ I V G i l , V G r = ∏ i ∈ I V G i r and V G = ∏ i ∈ I V G i , where G = ∏ i ∈ I G i endowed with the Tychonoff product topology. At the end section, we obtain that every compact strongly topological gyrogroup has property U and every Lindelof strongly topological gyrogroup has property ω-U.