Strongly Topological Gyrogroups with Remainders Close to Metrizable

Abstract

Some results between the properties of strongly topological gyrogroups and the properties of their remainders are established. In particular, if a strongly topological gyrogroup G is non-locally compact and G has a first-countable remainder, then $\chi (G)\le \omega _{1}$, $\omega (G)\le 2^{\omega }$ and $|bG|\le 2^{\omega _{1}}$. Moreover, it is proved that the property of paracompact p-space of a strongly topological gyrogroup G is equivalent with G having a Lindelöf remainder in a compactification. By this result, we prove that if H is a dense subspace of a strongly topological gyrogroup G which is locally pseudocompact and not locally compact, then every remainder of H is pseudocompact. Furthermore, if a strongly topological gyrogroup G has countable pseudocharacter and G is non-metrizable, then all remainders of G are pseudocompact. These two results give partial answers to a question posed by Arhangel’ skiǐ and Choban, see (Topol Appl 157:789–799, 2010, Problem 5.1). Finally, it is shown that the Lindelöf property of a non-locally compact strongly topological gyrogroup G is equivalent with having a remainder with subcountable type for some compactifications of G.