Advanced Search

A (Hausdorff) topological group is said to have a -base if it admits a base of neighbourhoods of the unit, $U_{\alpha }:\alpha \in {\mathbb{N}}^{\mathbb{N}}$, such that $U_{\alpha }\subset U_{\beta }$ whenever β ≤ α for all $\alpha ,\beta \in {\mathbb{N}}^{\mathbb{N}}$. 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...

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 ${(G,\tau )}^{\wedge }$ is a dense -closed subgroup of the compact group ${\left(G_d\right)}^{\wedge }$, where $G_d$ is the group G with the discrete...