On topological gyrogroups

Abstract

The concept of gyrogroups is a generalization of groups which do not explicitly have associativity. In this paper, we show that every first-countable strongly topological gyrogroup admits a left-invariant metric generating the original topology of it and every $T_0$ compact paratopological gyrogroup is a Hausdorff compact topological gyrogroup. Also, some basic properties on topological gyrogroups and paratopological gyrogroups are discussed.