On $\kappa$-Frechet-Urysohn topological groups

TL;DR

This paper characterizes $oldsymbol{ ext{kappa}- ext{Frechet-Urysohn}}$ topological groups, establishing conditions for hemicompact groups, subspace quotients, and products, and constructs a specific example under Martin's Axiom.

Contribution

It provides a new characterization of $oldsymbol{ ext{kappa}- ext{Frechet-Urysohn}}$ groups and explores their properties and examples within topological group theory.

Findings

Hemicompact groups are $oldsymbol{ ext{kappa}- ext{Frechet-Urysohn}}$ iff locally compact.

Quotients of certain topological vector spaces inherit the $oldsymbol{ ext{kappa}- ext{Frechet-Urysohn}}$ property.

Product of a $oldsymbol{ ext{kappa}- ext{Frechet-Urysohn}}$ tvs and a metrizable tvs is $oldsymbol{ ext{kappa}- ext{Frechet-Urysohn}}$.

Abstract

We characterize $\kappa$-Fr\'{e}chet--Urysohn topological groups. Using this characterization we show that: (1) a hemicompact topological group is $\kappa$-Fr\'{e}chet--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 $\kappa$-Fr\'{e}chet--Urysohn space, then also $E$ is a $\kappa$-Fr\'{e}chet--Urysohn space. Consequently, the product of a $\kappa$-Fr\'{e}chet--Urysohn tvs and a metrizable tvs is a $\kappa$-Fr\'{e}chet--Urysohn space. Under Martin's Axiom, we construct a countable Boolean $\kappa$-Fr\'{e}chet--Urysohn group which is not a $k_{\mathbb R}$-space.