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

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We dedicate this paper to Prof. Ofelia Alas on her 70th birthday. MSC:primary 54H11, 54B05 secondary 54E99A space X is called strongly pseudocompact if for each sequence (U n ) n∈N of pairwise disjoint nonempty open subsets of X there is a sequence (x n ) n∈N of points in X such that cl X ({x n : n ∈ N}) \ n∈N U n = ∅ and x n ∈ U n , for each n ∈ N. It is evident that every countably compact space is strongly pseudocompact and every strongly pseudocompact space is pseudocompact. In this paper, we construct a pseudocompact group that is not strongly pseudocompact answering two questions posed in [13].

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