Transversal, T1-independent, and T1-complementary paratopological group topologies

Abstract

We discuss the class of paratopological groups which admits a transversal, T1-independent and T1-complementary paratopological group topology. We show that the Sorgenfrey line does not admit a T1-complementary Hausdorff paratopological group topology, which gives a negative answer to [5, Problem 10]. We give a very useful criterion for transversality in term of submaximal paratopological group topology, and prove that if a non-discrete paratopological group topology G contains a central subgroup which admits a transversal paratopological group topology, then so does G. We introduce the concept of PT-sequence and give a characterization of an Abelian paratopological group being determined by a PT-sequence. As the applications, we prove that the Abelian paratopological group, which is endowed with the strongest paratopological group topology being determined by a T-sequence, does not admit a T1-complementary Hausdorff paratopological group topology on G. Finally, we study the class of countable paratopological groups which is determined by a PT-filter, and obtain a sufficient condition for a countable paratopological group G being determined by a PT-sequence which admits a transversal paratopological group topology on G being determined by a PT-sequence.