Some properties on topological gyrogroups

Abstract

In this paper, some properties of  topological gyrogroup are discussed. Among all, we show the following results: (1) It is shown that every first-countable left $\omega$-narrow semitopological gyrogroup is separable; (2) let $K$ a compact subset and $F$ a closed subset of a topological gyrogroup $G$ with $K\cap F=\emptyset$. There is an open neighborhood $V$ of $e$ in $G$ such that $(K\oplus V)\cap F=\emptyset$, and $(V\oplus K)\cap F=\emptyset$.  The two results answer the questions  \cite[Question 3.12]{BZX} and \cite[Question 4.6]{BSX}, respectively.