  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.
