Answer some Tkachenko's questions in factorizable semitopological groups

Abstract

We construct a Hausdorff quasitopological abelian group that is second-countable and semiregular, with a bounded continuous function that does not factor through any regular first-countable semitopological group. Its regular and Tychonoff reflections coincide, have a countable network, and fail even factorizability through arbitrary first-countable semitopological groups. This gives negative answers to \cite[Problems~6.1, 6.2, 6.4, and~6.5]{Tk2026} of Tkachenko. The construction starts with a countable cone group whose common separation reflection is not Fréchet--Urysohn, answering \cite[Problems~6.8 and~6.9]{Tk2026} negatively. In contrast, symmetrization gives affirmative answers to \cite[Problems~6.6 and~6.7]{Tk2026} without separation assumptions. We also show that every countable semitopological group admits simultaneous factorizations of countable families through second-countable Tychonoff semitopological groups.