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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 prove that the locally convex space C p (X) of continuous realvalued functions on a Tychonoff space X equipped with the topology of pointwise convergence is distinguished if and only if X is a Δ-space in the sense of Knight in [Trans. Amer. Math. Soc. 339 (1993), pp. 45-60]. As an application of this characterization theorem we obtain the following results: 1) If X is aČech-complete (in particular, compact) space such that C p (X) is distinguished, then X is scattered. 2) For every separable compact space of the Isbell-Mrówka type X, the space C p (X) is distinguished. 3) If X is the compact space of ordinals [0, ω 1 ], then C p (X) is not distinguished. We observe that the existence of an uncountable separable metrizable space X such that C p (X) is distinguished, is independent of ZFC. We also explore the question to which extent the class of Δ-spaces is invariant under basic topological operations.