On topological groups with a small base and metrizability

Saak Gabriyelyan ; Jerzy Kąkol ; Arkady Leiderman

Fundamenta Mathematicae, Tome 228 (2015), p. 129-158 / Harvested from The Polish Digital Mathematics Library

Résumé

A (Hausdorff) topological group is said to have a -base if it admits a base of neighbourhoods of the unit, ${U_{α}: α ∈ ℕ^{ℕ}}$, such that $U_{α} ⊂ U_{β}$ whenever β ≤ α for all $α, β ∈ ℕ^{ℕ}$. The class of all metrizable topological groups is a proper subclass of the class $TG_{}$ of all topological groups having a -base. We prove that a topological group is metrizable iff it is Fréchet-Urysohn and has a -base. We also show that any precompact set in a topological group $G ∈ TG_{}$ is metrizable, and hence G is strictly angelic. We deduce from this result that an almost metrizable group is metrizable iff it has a -base. Characterizations of metrizability of topological vector spaces, in particular of $C_{c}(X)$, are given using -bases. We prove that if X is a submetrizable $k_{ω}$-space, then the free abelian topological group A(X) and the free locally convex topological space L(X) have a -base. Another class $TG_{}$ of topological groups with a compact resolution swallowing compact sets appears naturally. We show that $TG_{}$ and $TG_{}$ are in some sense dual to each other. We conclude with a dozen open questions and various (counter)examples.

Détail

Publié le : 2015-01-01

Zbl 1334.22002

EUDML-ID : urn:eudml:doc:283157