On $κ$-Frechet--Urysohn topological groups
Saak Gabriyelyan, Alexander Osipov, Evgenii Reznichenko
TL;DR
The paper characterizes $\kappa$-Fréchet--Urysohn topological groups with a concrete sequence-based criterion, enabling new structural results. It establishes equivalences for hemicompact groups, showing $\kappa$-Fréchet--Urysohn, locally compact and Baire are all equivalent, and proves product stability with metrizable tvs. It also proves a quotient- lifting property for topological vector spaces and provides an MA-based construction of a countable Boolean $\kappa$-Fréchet--Urysohn group that is not a $k_\mathbb{R}$-space, answering a longstanding question in a stronger form. These results advance understanding of when $\kappa$-Fréchet--Urysohn behavior passes to products, quotients, and specific group constructions.
Abstract
We characterize $κ$-Fréchet--Urysohn topological groups. Using this characterization we show that: (1) a hemicompact topological group is $κ$-Fréchet--Urysohn iff it is locally compact, and (2) if $F$ is a closed metrizable subspace of a topological vector space (tvs) $E$ such that the quotient $E/F$ is a $κ$-Fréchet--Urysohn space, then also $E$ is a $κ$-Fréchet--Urysohn space. Consequently, the product of a $κ$-Fréchet--Urysohn tvs and a metrizable tvs is a $κ$-Fréchet--Urysohn space. Under Martin's Axiom, we construct a countable Boolean $κ$-Fréchet--Urysohn group which is not a $k_{\mathbb R}$-space.