On topological groups admitting a base at the identity indexed by $\omega ^{\omega }$
Volume 238 / 2017
Abstract
A topological group $G$ is said to have a local $\omega ^\omega $-base if the neighbourhood system at the 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 local $\omega ^\omega $-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-archimedean ordered fields lead to natural families of non-metrizable groups with a local $\omega ^\omega $-base which nevertheless are Baire topological spaces.
More examples come from such constructions as the free topological group $F(X)$ and the free Abelian topological group $A(X)$ of a Tychonoff (more generally uniform) space $X$, as well as the free product of topological groups. We show that 1) the free product of countably many separable topological groups with a local $\omega ^\omega $-base admits a local $\omega ^\omega $-base; 2) the group $A(X)$ of a Tychonoff space $X$ admits a local $\omega ^\omega $-base if and only if the finest uniformity of $X$ has an $\omega ^\omega $-base; 3) the group $F(X)$ of a Tychonoff space $X$ admits a local $\omega ^\omega $-base provided $X$ is separable and the finest uniformity of $X$ has an $\omega ^\omega $-base.
