Complete regularity of paratopological gyrogroups

The construction of Hartman-Mycielski in topological gyrogroups

“…If a triple (G, τ, ⊕) satisfies the first two conditions and its binary operation is continuous, we call such triple a paratopological gyrogroup [3]. Sometimes we will just say that G is a topological gyrogroup (paratopological gyrogroup) if the binary operation and the topology are clear from the context.…”

“…Atiponrat [2] extended the idea of topological groups to topological gyrogroups as gyrogroups with a topology such that its binary operation is jointly continuous and the operation of taking the inverse is continuous. Some basic properties of topological gyrogroups are studied in some detail; see, for instance [2,3,6].…”

On paratopological gyrogroups

“…x → ⊖x, is continuous. If a triple (G, τ, ⊕) satisfies the first two conditions and its binary operation is continuous, we call such triple a paratopological gyrogroup [3]. Sometimes we will just say that G is a topological gyrogroup (paratopo-logical gyrogroup) if the binary operation and the topology are clear from the context.…”

“…The Möbius gyrogroup D = {z ∈ C : |z| < 1} with the standard topology is a topological gyrogroup(see [3,Example 2]). In fact, D is micro-associative (see [3,Example 8]). Also, one can easily show that D is locally gyroscopic invariant.…”

“…Recently, Atiponrat and Maungchang [3] studied some separation axioms of paratopological gyrogroups. They proved that: Theorem 1.1.…”