Embedding of Strongly Topological Gyrogroups in Path-Connected and Locally Path-Connected Gyrogroups

Abstract

A gyrogroup, an algebraic structure that generalizes groups, is modeled on the bounded symmetric space of relativistically admissible velocities endowed with Einstein’s addition. Given a gyrogroup G, we offer a new way to construct a gyrogroup G• such that G• contains a gyro-isomorphic copy of G. We then prove that every strongly topological gyrogroup G can be embedded as a closed subgyrogroup of the path-connected and locally path-connected topological gyrogroup G•. We also study several properties shared by G and G•, including gyrocommutativity, first countability and metrizability. As an application of these results, we prove that being a quasitopological gyrogroup is equivalent to being a strongly topological gyrogroup in the class of normed gyrogroups.

Extension of Haar’s theorem

“…It seems that strongly topological gyrogroups suitably generalize topological groups. Several results that are valid for topological groups can be extended to the case of strongly topological gyrogroups in a natural way; see, for instance, [2,3,15,16]. This work is a continuation of the study of strongly topological gyrogroups.…”

Geometry of gyrogroups via Klein's approach

“…Proof. Let a, x ∈ G. By Lemma 1 of [18], L a (B(x, ε)) = a ⊕ B(x, ε) = B(a ⊕ x, ε). Hence, L a (B(x, ε)) ∈ B ε .…”

The construction of Hartman-Mycielski in topological gyrogroups

“…Since L f (g) and L ⊖g are homeomorphism, we get f is continuous and open at g. Proposition 3.4. [18] Let G be a gyrogroup. The function i Proof.…”

“…In 1958, Hartman and Mycielski proved that every Hausdorff topological group can be embedded into a Hausdorff path-connected, locally path-connected groups [11]. Recently, Wattanapan et al extend this result to strongly topological gyrogroups [18].…”

“…Construction of Hartman-Mycielski [18]: Let G be a gyrogroup with identity e and let J = [0, 1). A function f : J → G is a step function if there are real numbers a 0 , a 1 , ..., a n such that 0 = a 0 < a 1 < • • • < a n = 1 and f is constant on [a k , a k+1 ) for all k = 0, 1, ..., n − 1.…”