On topological groups with a small base and metrizability

A (Hausdorff) topological group is said to have a φ-base if it admits a base of neighbourhoods of the unit, {Uα : α ε N^N}, such that Uα ⊂ Uβ whenever β ≤ α for all α, β ε N^N. The class of all metrizable topological groups is a proper subclass of the class TGφ of all topological groups having a φ-base. We prove that a topological group is metrizable iff it is Fréchet-Urysohn and has a φ-base. We also show that any precompact set in a topological group G ε 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 φ-base. Characterizations of metrizability of topological vector spaces, in particular of Cc(X), are given using φ-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 φ-base. Another class TGCR of topological groups with a compact resolution swallowing compact sets appears naturally. We show that TGCR and TGG are in some sense dual to each other. We conclude with a dozen open questions and various (counter)examples.