Feathered gyrogroups and gyrogroups with countable pseudocharacter

TL;DR

This paper explores properties of feathered topological gyrogroups, establishing conditions for paracompactness, first-countability, and Raikov completeness, thus advancing understanding of their structure and answering open questions.

Contribution

It proves that feathered strongly topological gyrogroups are paracompact and D-spaces, and shows locally compact NSS-gyrogroups are first-countable, addressing open problems in the field.

Findings

Feathered strongly topological gyrogroups are paracompact.

Locally compact NSS-gyrogroups are first-countable.

Lindelöf P-gyrogroups are Raikov complete.

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' ski\v\i ~(2010) in \cite{AA1}. Moreover, we prove that every locally compact $NSS$-gyrogroup is first-countable. Finally, we prove that each Lindel\"{o}f $P$-gyrogroup is Ra$\check{\imath}$kov complete.