On topological groups admitting a base at identity indexed with ω^ω

Arkady G. Leiderman, Vladimir G. Pestov, Artur H. Tomita

Published 2015-11-22Version 1

A topological group $G$ is said to have a $\mathfrak G$-base if the neighbourhood system at identity admits a monotone cofinal map from the directed set $\omega^\omega$. In particular, every metrizable group is such, but the class of groups with a $\mathfrak G$-base is significantly wider. The aim of this article is to better understand the boundaries of this class, by presenting new examples and counter-examples. Ultraproducts and non-arichimedean ordered fields lead to natural families of non-metrizable groups with a $\mathfrak G$-base which nevertheless have the Baire property. More examples come from such constructions as the free topological group and the free Abelian topological group of a Tychonoff (more generally uniform) space, as well as the free product of topological groups. Our results answer some questions previously stated in the literature.