On quasitopological groups

In this paper, we prove that a quasitopological group is semi-metrizable if and only if it is first-countable. We also construct an example of quasitopological group to give a partial answer to Open Problem 4.6.9 in [A.V. Arhangelʼskii, M.G. Tkachenko, Topological Groups and Related Structures, Atlantis Press, World Scientific, 2008].