### abstract

- A (Hausdorff) topological group is said to have a G-base if it admits a base of neighbourhoods of the unit,{Uα: α∈ NN}, such that Uα⊂ Uβ whenever β≤ α for all α, β∈ NN. The class of all metrizable topological groups is a proper subclass of the class TGG of all topological groups having a G-base. We prove that a topological group is metrizable iff it is Fréchet–Urysohn and has a G-base. We also show that any precompact set in a topological group G∈ TGG is metrizable, and hence G is strictly angelic. We deduce from this result that an almost metrizable group is metrizable iff it has a G-base. Characterizations of metrizability of topological vector spaces, in particular of Cc (X), are given using G-bases. We prove that if X is a submetrizable kω-space, then the free abelian topological group A (X) and the free locally convex topological space L (X) have a G-base. Another class TGCR …