On paratopological groups

In this paper, we firstly construct a Hausdorff non-submetrizable paratopological group G in which every point is a G δ -set, which gives a negative answer to Arhangelʼskiı̌ and Tkachenkoʼs question [A.V. Arhangelʼskiı̌, M. Tkachenko, Topological Groups and Related Structures, Atlantis Press and World Sci., 2008]. We also prove that each first-countable Abelian paratopological group is submetrizable. Moreover, we discuss developable paratopological groups and construct a non-metrizable, separable, Moore paratopological group. Further, we prove that a regular, countable, locally k ω -paratopological group is a discrete topological group or contains a closed copy of S ω . Finally, we discuss some properties on non-H-closed paratopological groups, and show that Sorgenfrey line is not H-closed, which gives a negative answer to Arhangelʼskiı̌ and Tkachenkoʼs question [A.V. Arhangelʼskiı̌, M. Tkachenko, Topological Groups and Related Structures, Atlantis Press and World Sci., 2008]. Some questions are posed.