varkappa-Fréchet--Urysohn and Baire subgroups of free topological groups
Abstract
A topological space is varkappa-Fréchet--Urysohn if every point in the closure of an open set U is the limit of a sequence of points of U. We prove that every varkappa-Fréchet--Urysohn subgroup and every Baire subgroup of the free topological group on a Tychonoff space X in any full variety of topological groups (in particular, of the free topological group F(X), the free Abelian topological group A(X), and the free Boolean topological group B(X)) is discrete. In particular, each of these groups is varkappa-Fréchet--Urysohn (or Baire) only if X is discrete, and all their metrizable and even Fréchet--Urysohn subgroups are discrete. Along the way we characterize strongly varkappa-Fréchet--Urysohn spaces in the spirit of Michael's theorem and study countable families of closed sets absorbing convergent sequences in varkappa-Fréchet--Urysohn spaces and groups.
Related
- Metrizable and Fréchet--Urysohn subgroups and linear subspaces of topological vector spaces
- Stabilizers and maximal subgroups of topological full groups
- Maximal subgroups of topological full groups arising from factor maps and subgroupoids
- Maximal subgroups of topological full groups of shifts of finite type
- Finite abelian subgroups of algebraic groups
BibTeX
@misc{leiderman2026varkappafrécheturysohn,
title = {{$\varkappa$-Fréchet--Urysohn and Baire subgroups of free topological groups}},
author = {Arkady Leiderman and Evgenii Reznichenko and Ol'ga Sipacheva},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.07401},
eprint = {2610.07401},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.07401},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - varkappa-Fréchet--Urysohn and Baire subgroups of free topological groups
AU - Arkady Leiderman
AU - Evgenii Reznichenko
AU - Ol'ga Sipacheva
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.07401
UR - https://arxiv.org/abs/2610.07401
AB - A topological space is varkappa-Fréchet--Urysohn if every point in the closure of an open set U is the limit of a sequence of points of U. We prove that every varkappa-Fréchet--Urysohn subgroup and every Baire subgroup of the free topological group on a Tychonoff space X in any full variety of topological groups (in particular, of the free topological group F(X), the free Abelian topological group A(X), and the free Boolean topological group B(X)) is discrete. In particular, each of these groups is varkappa-Fréchet--Urysohn (or Baire) only if X is discrete, and all their metrizable and even Fréchet--Urysohn subgroups are discrete. Along the way we characterize strongly varkappa-Fréchet--Urysohn spaces in the spirit of Michael's theorem and study countable families of closed sets absorbing convergent sequences in varkappa-Fréchet--Urysohn spaces and groups.
ER -