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On paratopological groups
In this paper, we firstly construct a Hausdorff non-submetrizable
paratopological group in which every point is a -set, which
gives a negative answer to Arhangel'ski\v{\i}\ and Tkachenko's question
[Topological Groups and Related Structures, Atlantis Press and World Sci.,
2008]. We prove that each first-countable Abelian paratopological group is
submetrizable. Moreover, we discuss developable paratopological groups and
construct a non-metrizable, Moore paratopological group. Further, we prove that
a regular, countable, locally -paratopological group is a discrete
topological group or contains a closed copy of . 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
Arhangel'ski\v{\i}\ and Tkachenko's question [Topological Groups and Related
Structures, Atlantis Press and World Sci., 2008]. Some questions are posed.Comment: 14 page
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