Separately continuous maps into regular spaces and continuity of operations in separable semitopological groups
Abstract
A space is feebly compact if every locally finite family of nonempty open subsets of it is finite; a Tychonoff space is feebly compact if and only if it is pseudocompact. M. Tkachenko asked whether a regular feebly compact semitopological group is a topological group provided that it is separable, has countable tightness, or is a k-space; for Tychonoff groups the answer is known to be positive. We answer this question in the separable case: every regular separable feebly compact semitopological group is a topological group. The proof is based on the following result: if X is a separable Baire space, Y is a feebly compact space and Z is a regular space, then every separately continuous map X× Y→ Z is strongly quasicontinuous. Unlike the known approaches for Tychonoff spaces, the proof does not use real-valued functions. More generally, every regular separable semitopological group which is ΓBMo-nonmeager, a Baire-type property defined by a modification of the Banach-Mazur game, is a topological group. This class of groups contains all regular separable Baire groups which are feebly p-spaces; feebly p-spaces form a wider class of spaces, which contains all feebly compact spaces and all p-spaces. The results remain valid for spaces satisfying the assumptions locally. In particular, every regular locally separable locally feebly compact semitopological group is a topological group.
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BibTeX
@misc{reznichenko2026separately,
title = {{Separately continuous maps into regular spaces and continuity of operations in separable semitopological groups}},
author = {Evgenii Reznichenko},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.04891},
eprint = {2611.04891},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.04891},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Separately continuous maps into regular spaces and continuity of operations in separable semitopological groups
AU - Evgenii Reznichenko
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.04891
UR - https://arxiv.org/abs/2611.04891
AB - A space is feebly compact if every locally finite family of nonempty open subsets of it is finite; a Tychonoff space is feebly compact if and only if it is pseudocompact. M. Tkachenko asked whether a regular feebly compact semitopological group is a topological group provided that it is separable, has countable tightness, or is a k-space; for Tychonoff groups the answer is known to be positive. We answer this question in the separable case: every regular separable feebly compact semitopological group is a topological group. The proof is based on the following result: if X is a separable Baire space, Y is a feebly compact space and Z is a regular space, then every separately continuous map X× Y→ Z is strongly quasicontinuous. Unlike the known approaches for Tychonoff spaces, the proof does not use real-valued functions. More generally, every regular separable semitopological group which is Γ^BM_o-nonmeager, a Baire-type property defined by a modification of the Banach-Mazur game, is a topological group. This class of groups contains all regular separable Baire groups which are feebly p-spaces; feebly p-spaces form a wider class of spaces, which contains all feebly compact spaces and all p-spaces. The results remain valid for spaces satisfying the assumptions locally. In particular, every regular locally separable locally feebly compact semitopological group is a topological group.
ER -