On $\hat{I}º$-Frechet-Urysohn topological groups

Abstract

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