A special class of semi(quasi)topological groups and three-space properties

The multiplication of a semitopological (quasitopological) group G is called sequentially continuous if the product map of G×G into G is sequentially continuous. In this paper, we mainly consider the properties of semitopological (quasitopological) groups with sequentially continuous multiplications and three-space problems in quasitopological groups. It is showed that (1) every snf-countable semitopological group G with the sequentially continuous multiplication is sof-countable; (2) if G is a sequential quasitopological group with the sequentially continuous multiplication, then G contains a closed copy of Sω if and only if it contains a closed copy of S2, which give a partial answer to a problem posed by R.-X. Shen; (3) let G be a quasitopological group with the sequentially continuous multiplication, then the following are equivalent: (i) G is a sequential α4-space; (ii) G is Fréchet; (iii) G is strongly Fréchet; (4) (MA+¬CH) there exists a non-metrizable, separable, normal and Moore quasitopological group; (5) some examples are constructed to show that metrizability, first-countability and second-countability are not three-space properties in the class of quasitopological groups.