Feathered gyrogroups and gyrogroups with countable pseudocharacter

Abstract

Topological gyrogroups, with a weaker algebraic structure than groups, have been investigated recently. In this paper, we prove that every feathered strongly topological gyrogroup is paracompact, which implies that every feathered strongly topological gyrogroup is a D-space and gives partial answers to two questions posed by A.V.Arhangel' skiǐ (2010) in [2]. Moreover, we prove that every locally compact NSS-gyrogroup is first-countable. Finally, we prove that each Lindelöf P-gyrogroup is Raǐkov complete.

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.…”

“…Suppose that G is a topological gyrogroup, and let A, B be subsets of G. If A and B are compact, then A ⊕ B is compact. Definition 2.3 (p. 5116, [2]). A topological gyrogroup G is strong if there exists a neighborhood base U at the identity e of G such that gyr[x, y](U ) = U for all x, y ∈ G, U ∈ U.…”

“…In [1], W. Atiponrat introduced the notion of a topological gyrogroup, a gyrogroup with a compatible topology. In [2], M. Bao and F. Lin defined the notion of a strongly topological gyrogroup. It seems that strongly topological gyrogroups suitably generalize topological groups.…”

Separability in (strongly) topological gyrogroups

“…Definition 2.5. ( [3]) Let G be a topological gyrogroup. We say that G is a strongly topological gyrogroup if there exists a neighborhood base U of 0 such that, for every U ∈ U , gyr[x, y](U) = U for any x, y ∈ G. For convenience, we say that G is a strongly topological gyrogroup with neighborhood base U of 0.…”

“…Then Z. Cai, S. Lin and W. He in [7] proved that every topological gyrogroup is a rectifiable space. In 2019, the authors [3] defined the concept of strongly topological gyrogroups and found that M öbius gyrogroups, Einstein gyrogroups, and Proper Velocity gyrogroups are all strongly topological gyrogroups. Furthermore, the authors gave a characterization for a strongly topological gyrogroup being a feathered space, that is, a strongly topological gyrogroup G is feathered if and only if it contains a compact L-subgyrogroup H such that the quotient space G/H is metrizable.…”