On topological groups admitting a base at the identity indexed by w ^w

A topological group G is said to have a local ^w-base if the neighbourhood system at the identity admits a monotone cofinal map from the directed set ^w. In particular, every metrizable group is such, but the class of groups with a local ^w-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 nonarchimedean ordered fields lead to natural families of non-metrizable groups with a local ^w-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 ^w-base admits a local ^w-base; 2) the group A(X) of a Tychonoff space X admits a local ^w-base if and only if the finest uniformity of X has an ^w-base; 3) the group F(X) of a Tychonoff space X admits a local ^w-base provided X is separable and the finest uniformity of X has an ^w-base.