Characterizing sequences for precompact group topologies

Abstract

A precompact group topology τ on an abelian group G is called single sequence characterized (for short, ss-characterized) if there is a sequence u = (un) in G such that τ is the finest precompact group topology on G making u = (un) converge to zero. It is proved that a metrizable precompact abelian group (G, τ ) is ss-characterized iff it is countable. For every metrizable precompact group topology τ on a countably infinite abelian group G there exists a group topology η such that η is strictly finer than τ and the groups (G, τ ) and (G, η) have the equal Pontryagin dual groups. We give a complete description of all ss-characterized precompact abelian groups modulo countable ss-characterized groups from which we derive:(1) No infinite pseudocompact abelian group is ss-characterized.(2) An ss-characterized precompact abelian group is hereditarily disconnected.

Characterized subgroups of the circle group

“…Therefore, T tu(T) is the finest totally bounded group topology τ on Z with u n τ − → 0. Here we list some properties of this topology (see also [32] for further properties): One can find in [21] many examples of totally bounded group topologies τ = T H on Z without any non-trivial convergent sequence. To this end one has to choose the subgroup H of T such that u n T H − − → 0 (i.e., H ≤ t u (T)) never occurs for a non-trivial sequence u in Z. Non-measurable subgroups H of T satisfy this condition (for more details see [21] and [73]).…”

“…For other results on characterized subgroups and related topics see the recent papers [38,31,32,33,35,36,37,52,53,54,55,70].…”

On reflexivity of the group of the null sequences valued in an Abelian topological group

“…In particular, for every T -sequence u in a group G the group (G, τ u ) is an s-group. The classes of s-and bs-groups are thoroughly studied in [15,16] (see also [8]). Note that every sequential group is an s-group [15].…”

Precompact groups and convergence

“…We show that in ZFC, some natural 'minimal' precompact topologies on countable groups are sequential if an only if they are metrizable (Corollary 1), answering a question of D. Dikranjan et. al in [5]. We use the effective topology techniques pioneered by S. Todorčevic (see [18] and [19]) to study such groups.…”