Some properties of semicompletions of paratopological groups

Abstract

We study the Raĭkov completion Gˇ of a paratopological group G defined by Banakh-Ravsky in 2020. This permits to Banakh and Tkachenko introduce the notion of C-semicompletion, where C is a class of continuous homomorphisms of paratopological groups. We show that the product of C-semicompletions is a C-semicompletion for the product of arbitrary paratopological groups. In particular, if we consider the class C of all continuous homomorphisms between paratopological groups, we obtain that Gˇ=∏i∈IGˇi. We also study the interaction between C-semicompletions of a paratopological group G and the C-semicompletions of subgroups of G.We prove that if G is a T3 paratopological group, then every C-semicompletion of G is T3. We show that if G is first-countable (second-countable), then every C-semicompletion of G is first-countable (second-countable). Therefore, if G is metrizable separable, then every C-semicompletion of G is metrizable separable as well.