Three-space properties in paratopological gyrogroups

Abstract

In this paper, we consider the three-space properties in paratopological gyrogroups. The following are the established conclusions:
(1) metrizability of compact (resp., sequentially compact, countably compact) subsets is a three-space
property in the class of $k$-gentle paratopological gyrogroups;
(2) let $G$ be a strongly paratopological gyrocommutative gyrogroup and let $H$ be a second-countable invariant topological subgyrogroup of $G$. If the paratopological gyrogroup $G/H$ has a countable network, then so does $G$;
(3) let $H$ be a compact strongly $L$-subgyrogroup of a paratopological gyrogroup $G$. If $H$ and $G/H$ have
countable tightness, then $G$ has countable tightness.