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Currently displaying 1 – 2 of 2

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On topological groups with a small base and metrizability

Saak Gabriyelyan; Jerzy Kąkol; Arkady Leiderman — 2015

Fundamenta Mathematicae

A (Hausdorff) topological group is said to have a -base if it admits a base of neighbourhoods of the unit, $U_{α} : α \in ℕ^{ℕ}$ , such that $U_{α} \subset U_{β}$ whenever β ≤ α for all $α, β \in ℕ^{ℕ}$ . The class of all metrizable topological groups is a proper subclass of the class $T G_{}$ 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 $G \in T G_{}$ is metrizable, and hence G is strictly angelic. We deduce from this result...

On a generalization of Abelian sequential groups

Saak S. Gabriyelyan — 2013

Fundamenta Mathematicae

Let (G,τ) be a Hausdorff Abelian topological group. It is called an s-group (resp. a bs-group) if there is a set S of sequences in G such that τ is the finest Hausdorff (resp. precompact) group topology on G in which every sequence of S converges to zero. Characterizations of Abelian s- and bs-groups are given. If (G,τ) is a maximally almost periodic (MAP) Abelian s-group, then its Pontryagin dual group ${(G, τ)}^{\land}$ is a dense -closed subgroup of the compact group ${(G_{d})}^{\land}$ , where $G_{d}$ is the group G with the discrete...

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