Non metrizable topologies on Z with countable dual group.

Daniel de la Barrera Mayoral

doi:10.4995/agt.2017.4469

Non metrizable topologies on Z with countable dual group.

Daniel de la Barrera Mayoral

DOI: https://doi.org/10.4995/agt.2017.4469

Abstract

In this paper we give two families of non-metrizable topologies on the group of the integers having a countable dual group which is isomorphic to a infinite torsion subgroup of the unit circle in the complex plane. Both families are related to D-sequences, which are sequences of natural numbers such that each term divides the following. The first family consists of locally quasi-convex group topologies. The second consists of complete topologies which are not locally quasi-convex. In order to study the dual groups for both families we need to make numerical considerations of independent interest.

Keywords

Locally quasi-convex topology; D-sequence; continuous character; infinite torsion subgroups of T.

Subject classification

2010 Mathematics Subject Classification

Primary: 22A05, 55M05. Secondary: 20K45.

Full Text:

PDF

References

L. Außenhofer, Contributions to the duality theory of abelian topological groups and to the theory of nuclear groups, Dissertationes Math. (Rozprawy Mat.) 384, (1999).

L. Außenhofer and D. de la Barrera Mayoral, Linear topologies on Z are not Mackey topologies, Journal of Pure and Applied Algebra 216, no. 6 (2012), 1340-1347.

https://doi.org/10.1016/j.jpaa.2012.01.005

L. Außenhofer, D. de la Barrera Mayoral, D. Dikranjan and E. Martín-Peinador, 'Varopoulos paradigm': Mackey property versus metrizability in topological groups, Revista Matemática Complutense 30, no. 1 (2017), 167-184.

https://doi.org/10.1007/s13163-016-0209-y

L. Außenhofer, D. Dikranjan and E. Martín-Peinador, Locally quasi-convex compatible topologies on a topological group, Axioms 4 (2015), 436-458.

https://doi.org/10.3390/axioms4040436

D. de la Barrera, Duality on abelian topological groups: The Mackey Problem, PhD Thesis, Universidad Complutense de Madrid, (2015).

D. de la Barrera Mayoral, Q is not Mackey group, Topology Appl. 178 (2014), 265-275.

https://doi.org/10.1016/j.topol.2014.10.004

M. J. Chasco, Pontryagin duality for metrizable groups, Arch. Math. (Basel) 70, no. 1 (1998), 22-28.

https://doi.org/10.1007/s000130050160

M. J. Chasco, E. Martín-Peinador and V. Tarieladze, On Mackey topology for groups, Stud. Math. 132, no. 3 (1999), 257-284.

L. de Leo, Weak and strong topologies in topological abelian groups, PhD Thesis, Universidad Complutense de Madrid, July 2008.

J. M. Díaz Nieto and E. Martín-Peinador, Characteristics of the Mackey topology for Abelian topological groups. J. C. Ferrando and M. López-Pellicer (eds.), Descriptive Topology and Functional Analysis, Springer Proceedings in Mathematics & Statistics 80 .

https://doi.org/10.1007/978-3-319-05224-3_7

D. Dikranjan, S. S. Gabriyelyan and V. Tarieladze, Characterizing sequences for precompact group topologies, J. Math Anal. Appl. 412 (2014), 505-519.

https://doi.org/10.1016/j.jmaa.2013.10.047

D. Dikranjan, E. Martín-Peinador and V. Tarieladze, A class of metrizable locally quasi-convex groups which are not Mackey, Forum Math. 26 (2014), 723 -757

S. S. Gabriyelyan, Groups of quasi-invariance and the Pontryagin duality, Topology Appl. 157, no. 18, (2010), 2786-2802.

https://doi.org/10.1016/j.topol.2010.08.018

E. Martín-Peinador and V. Tarieladze, Mackey topology on locally convex spaces and on locally quasi-convex groups. Similarities and historical remarks, RACSAM 110, no. 2 (2016), 667-679.

https://doi.org/10.1007/s13398-015-0256-0

I. Protasov and E. Zelenyuk, Topologies on abelian groups, Math. USSR Izvestiya, 37, no. 2 (1991).

I. Protasov and E. Zelenyuk, Topologies on groups determined by sequences, Mathematical Studies, Monograph Series Vol. 4, Lviv, 1999.

N. Ya. Vilenkin, The theory of characters of topological Abelian groups with boundedness given, Izvestiya Akad. Nauk SSSR. Ser. Mat. 15 (1951), 439-162.

Abstract Views

994

Metrics Loading ...

_{Metrics powered by PLOS ALM}