The metrization problem for Fréchet groups

Moore, Justin Tatch; Todorčević, Stevo

doi:10.1016/b978-044452208-5/50023-5

Open Problems in Topology II 2007

DOI: 10.1016/b978-044452208-5/50023-5

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The metrization problem for Fréchet groups

Justin Tatch Moore¹,

Stevo Todorčević²

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Cited by 6 publications

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Characterizing sequences for precompact group topologies

Dikranjan

Gabriyelyan

Tarieladze

2014

Journal of Mathematical Analysis and Applications

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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.

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Characterizing sequences for precompact group topologies

Dikranjan

Gabriyelyan

Tarieladze

2014

Journal of Mathematical Analysis and Applications

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Precompact Fréchet topologies on Abelian groups

Hrušák¹,

Ramos-García²

2012

Topology and its Applications

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Cardinal invariants and convergence properties of locally minimal groups

Dikranjan

Shakhmatov

2020

Topology and its Applications

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If G is a locally essential subgroup of a compact abelian group K, then:Items (i)-(iii) hold when G is a dense locally minimal subgroup of K. We show that locally minimal locally precompact abelian groups of countable tightness are metrizable. In particular, a minimal abelian group of countable tightness is metrizable. This answers a question of O. Okunev posed in 2007.For every uncountable cardinal κ, we construct a Fréchet-Urysohn minimal group G of character κ such that the connected component of G is an open normal ω-bounded subgroup (thus, G is locally precompact). We also build a minimal nilpotent group of nilpotency class 2 without non-trivial convergent sequences having an open normal countably compact subgroup.All topological groups are assumed to be Hausdorff.

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The metrization problem for Fréchet groups

Cited by 6 publications

References 10 publications

Characterizing sequences for precompact group topologies

Characterizing sequences for precompact group topologies

Precompact Fréchet topologies on Abelian groups

Cardinal invariants and convergence properties of locally minimal groups

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