Every group admits a bad topology

Ajtai, Miklós; Havas, I.; Komlós, János

doi:10.1007/978-3-0348-5438-2_3

“…topology which do not for any n < ω admit an SSGP(n) topology. Indeed from Corollary 3.14 and Theorem 3.9 respectively we see that the groups G = Z and G = Z(p ∞ ) admit no SSGP(n) topology; while Ajtai, Havas, and Komlós [1], and later Zelenyuk and Protasov [41], have shown the existence of m.a.p. topologies for Z and for Z(p ∞ ).…”

Section: Definition 32mentioning

confidence: 99%

“…topology. (d) Ajtai, Havas and Komlós [1] proved that each group G of the form Z, Z(p ∞ ), or n Z(p n ) (with all p n ∈ P either identical or distinct) admits a m.a.p. group topology.…”

Section: (2332) (C)mentioning

confidence: 99%

Topologies on Abelian Groups

Zelenyuk¹,

1991

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A Hausdorff topological group G = (G, T ) has the small subgroup generating property (briefly: has the SSGP property, or is an SSGP group) if for each neighborhood U of 1G there is a family H of subgroups of G such that H ⊆ U and H is dense in G. The class of SSGP groups is defined and investigated with respect to the properties usually studied by topologists (products, quotients, passage to dense subgroups, and the like), and with respect to the familiar class of minimally almost periodic groups (the m.a.p. groups). Additional classes SSGP(n) for n < ω (with SSGP(1) = SSGP) are defined and investigated, and the class-theoretic inclusionsare established and shown proper. In passing the authors also establish the presence of SSGP(1) or SSGP(2) in many of the early examples in the literature of abelian m.a.p. groups.2010 MSC: Primary 54H11; Secondary 22A05.

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“…Now, define (algebraically) G n := H ⊕ G n−1 ; we give G n a metric topology T n which is different from the product topology, using a technique taken from M. Ajtai, I. Havas, and J. Komlós [1]. We create a metric group topology on G n starting with a function ν : S → R + , where S is a specified generating set for G n with 0 / ∈ S. S will typically be highly redundant as a generating set.…”

Section: Theorem 42 Prodanov's Group (G T ) Satisfies (G T ) ∈ Ssmentioning

confidence: 99%

Some classes of minimally almost periodic topological groups

Comfort

¹

,

Gould

²

2015

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A Hausdorff topological group G = (G, T ) has the small subgroup generating property (briefly: has the SSGP property, or is an SSGP group) if for each neighborhood U of 1G there is a family H of subgroups of G such that H ⊆ U and H is dense in G. The class of SSGP groups is defined and investigated with respect to the properties usually studied by topologists (products, quotients, passage to dense subgroups, and the like), and with respect to the familiar class of minimally almost periodic groups (the m.a.p. groups). Additional classes SSGP(n) for n < ω (with SSGP(1) = SSGP) are defined and investigated, and the class-theoretic inclusionsare established and shown proper. In passing the authors also establish the presence of SSGP(1) or SSGP(2) in many of the early examples in the literature of abelian m.a.p. groups.2010 MSC: Primary 54H11; Secondary 22A05.

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“…There are many examples of MAP monothetic Polish groups (see e.g. [2]). The following problem was posed in [12].…”

Section: Theorem (A-g)mentioning

confidence: 99%

Classifying dynamical systems by their recurrence properties

Glasner¹

2004

TMNA

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Abstract. In his seminal paper of 1967 on disjointness in topological dynamics and ergodic theory H. Furstenberg started a systematic study of transitive dynamical systems. In recent years this work served as a basis for a broad classification of dynamical systems by their recurrence properties. In this paper I describe some aspects of this new theory and its connections with combinatorics, harmonic analysis and the theory of topological groups.

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Every group admits a bad topology

Cited by 24 publications

References 1 publication

Topologies on Abelian Groups

Topologies on Abelian Groups

Some classes of minimally almost periodic topological groups

Classifying dynamical systems by their recurrence properties

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