On topological groups admitting a base at the identity indexed by $\omega ^{\omega }$

Abstract

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.

ω-dominated function spaces and ω-bases in free objects of topological algebra

“…Unifying Theorems 8.1 and 8.2 and taking into account that (ω ω ) ω ∼ = ω ω , we get the following characterization, whose "abelian" part was found also in [25], [30], [32]. Theorem 8.3.…”

“…It was shown earlier that (a) for a cosmic k ω -space the free objects A(X), F(X), L(X) have ω ω -bases [23], [20], [21], [31]. (b) for a metrizable σ-compact space X the free locally convex space L(X) has an ω ω -base [21]; 1 (c) the free Abelian topological group A(X) of a Tychonoff space X has an ω ω -base if and only if the universal uniformity U X of X has an ω ω -base [30], [32], [25]; (d) a uniform space X is ω ω -based iff the free Abelian topological group A u (X) has an ω ω -base [30]; (e) a separable uniform space X is ω ω -based iff the free topological group F u (X) has an ω ω -base [30]. In this paper we shall characterize uniform spaces whose free topological groups and free (locally convex) topological vector spaces have ω ω -bases.…”

“…The following description of the topology of a strong topological group over a uniform space is a small improvement of the known description of the topology of a free topological group, due to V. Pestov [34] (see also [36], [39], [30]). Theorem 7.2.…”

“…For the existence of such ordered field F , see e.g. [30]. Replacing F by the subfield generated by the set {ε(α)} α∈κ , we can additionally assume that |F | = κ.…”

Countable tightness and $${\mathfrak {G}}$$-bases on free topological groups

“…Recently, A.G. Leiderman, V.G. Pestov and A.H. Tomita in [26] showed the following two results: [26] Let X be a metrizable space and the set of all non-isolated points of X is a σ-compact subset of X. Then A(X) has a G-base.…”