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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We extend van Mill-Wattel's results and show that each countably compact completely regular space with a continuous selection on couples is suborderable. The result extends also to pseudocompact spaces if they are either scattered, first countable, or connected. An infinite pseudocompact topological group with such a continuous selection is homeomorphic to the Cantor set. A zero-selection is a selection on the hyperspace of closed sets which chooses always an isolated point of a set. Extending Fujii-Nogura results, we show that an almost compact space with a continuous zero-selection is homeomorphic to some ordinal space, and a (locally compact) pseudocompact space with a continuous zero-selection is an (open) subspace of some space of ordinals. Under the Diamond Principle, we construct several counterexamples, e.g. a locally compact locally countable monotonically normal space with a continuous zero-selection which is not suborderable.
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