On generalized metrizable properties in quasitopological groups

In this paper, we mainly discuss some generalized metrizable properties in quasitopological groups and topological groups. We show that every weakly first-countable quasitopological group is symmetrizable. As an application, we prove that the Sorgenfrey line is not homeomorphic to any quasitopological group, which gives a negative answer to Arhangelʹskiı̌ and Tkachenkoʹs [4, Open Problem 3.3.11]. Also we answer two questions on quasitopological groups posed by F. Lin in [10], and discuss the csf-countable topological groups.