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.
Related
- Separately continuous maps into regular spaces and continuity of operations in separable semitopological groups
- The answer about Itzkowitz's Problems on FSIN groups
- A positive answer to Barsotti and Carman's question on unit groups of Burnside rings of fusion systems
- Reflective algebras over factorizable Hopf algebras
- Methods to Find Integer Points on the Elliptic Curve of Factorizable Polynomial Equation
BibTeX
@misc{xie2026answer,
title = {{Answer some Tkachenko's questions in factorizable semitopological groups}},
author = {Li-Hong Xie},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15385},
eprint = {2610.15385},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15385},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Answer some Tkachenko's questions in factorizable semitopological groups
AU - Li-Hong Xie
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.15385
UR - https://arxiv.org/abs/2610.15385
AB - 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.
ER -