A note on (strongly) topological gyrogroups

Abstract

In this article, some properties of rectifiable spaces and (strongly) topological gyrogroups are investigated. It is proved that all Fréchet-Urysohn rectifiable spaces with countable cs⁎-character are metrizable, which deduces that all Fréchet-Urysohn topological gyrogroups with countable cs⁎-character are metrizable. Therefore, the question posed in [9, Problem 3.9] is answered affirmatively. More important, we show that if G is a strongly topological gyrogroup, H is a closed strong subgyrogroup of G and H is inner neutral, then G/H is first-countable if and only if G/H is Fréchet-Urysohn with an ωω-base. It is also shown that if G is a left ω-balanced strongly topological gyrogroup, there exists a continuous pseudometric d on G such that d(0,x⊕y)≤d(0,x)+d(0,y) and there is a closed left invariant strong subgyrogroup Z in G such that the quotient space G/Z is metrizable. Additionally, if G is an ω-balanced strongly topological gyrogroup and the subgyrogroup Z is left-constant, the quotient space G/Z is a metrizable topological gyrogroup.