Uniformities on strongly topological gyrogroups

摘要

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, circle plus, tau). 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,V-H = V-H; (3) if {(G(i), circle plus(i), tau(i)) : i is an element of I} is a family of strongly topological gyrogroups, then V-G(l) = Pi V-i is an element of I(Gi)l, V-G(r) = Pi V-i is an element of I(Gi)r and V-G = Pi V-i is an element of I(Gi), where G = Pi(i is an element of 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 omega-U. (c) 2021 Elsevier B.V. All rights reserved.