We construct a precompact completely regular paratopological
Abelian group G of size (2ω)+ such that all subsets
of G of cardinality ≤ 2ω are closed. This shows that Protasov’s
theorem on non-closed discrete subsets of precompact
topological groups cannot be extended to paratopological
groups. We also prove that the group reflection of the
product of an arbitrary family of paratopological (even semitopological)
groups is topologically isomorphic to the product
of the group reflections of the factors, and that the group reflection,
H*, of a dense subgroup G of a paratopological group
G is topologically isomorphic to a subgroup of G*.
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