Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Institution: Lomonosov Moscow State University
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Russian Science Foundation
Awards: G7787288036, G8893505604
OpenAlex ID: W7220894997