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In this paper, we firstly construct a Hausdorff non-submetrizable paratopological group G in which every point is a Gδ-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...
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