Group reflection and precompact paratopological groups

Mikhail Tkachenko 1
  • 1 Departamento de Matemáticas, Universidad Autónoma Metropolitana Av. San Rafael Atlixco 186, Col. Vicentina, Iztapalapa C.P. 09340, México, D.F., Mexico


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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