Characterizable groups: Some results and open questions

Topology and its Applications

Volumn 159, Issue 9, 2012, Pages 2378-2391

  Gabriyelyan, S S ^a  

^a BEN GURION UNIVERSITY OF THE NEGEV   (Israel)

Author keywords

Characterizable group; Characterized subgroup; G closed subgroup; Polish group; T sequence; TB sequence; Topologically torsion element

Indexed keywords

EID: 84860837528     PISSN: 01668641     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.topol.2011.11.060     Document Type: Article

References (43)

1. Armacost D.L. The Structure of Locally Compact Abelian Groups. Monogr. Textb. Pure Appl. Math. 1981, vol. 68. Marcel Dekker, Inc., New York.
2. Außenhofer L. Contributions to the duality theory of Abelian topological groups and to the theory of nuclear groups. Dissertationes Math. (Rozprawy Mat.) 1999, 384. Dissertation, Tübingen, 1998, 113 p.
3. Banaszczyk W., Chasco M.J., Martin-Peinador E. Open subgroups and Pontryagin duality. Math. Z. 1994, 215:195-204.
4. Barbieri G., Dikranjan D., Milan C., Weber H. Answer to Raczkowski's question on convergent sequences of integers. Topology Appl. 2003, 132:89-101.
5. Barbieri G., Dikranjan D., Milan C., Weber H. Convergent sequences in precompact group topologies. Appl. Gen. Topol. 2005, 6(2):149-169.
6. Barbieri G., Dikranjan D., Milan C., Weber H. Topological torsion related to some sequences of integers. Math. Nachr. 2008, 281(7):930-950.
7. Beiglböck M., Steineder C., Winkler R. Sequences and filters of characters characterizing subgroups of compact Abelian groups. Topology Appl. 2006, 153:1682-1695.
8. Biró A. Strong characterizing sequences for subgroups of compact groups. J. Number Theory 2006, 121:324-354.
9. Biró A. Characterizations of groups generated by Kronecker sets. J. Theor. Nombres Bordeaux 2007, 19:567-582.
10. Biró A., Deshouillers J.-M., Sós V.T. Good approximation and characterization of subgroups of R/Z. Studia Sci. Math. Hungar. 2001, 38:97-113.
11. Biró A., Sós V.T. Strong characterizing sequences in simultaneous Diophantine approximation. J. Number Theory 2003, 99:405-414.
12. Borel J.-P. Sous-groupes de R liés à la répartition modulo 1 de suites. Ann. Fac. Sci. Toulouse 1983, 5:218-235.
13. Borel J.-P. Sur certains sous-groupes de R liés à la suite des factorielles. Colloq. Math. 1991, 62(1):21-30.
14. Braconnier J. Sur les groupes topologiques primaires. C. R. Acad. Sci. Paris 1944, 218:304-305.
15. Chasco M.J. Pontryagin duality for metrizable groups. Arch. Math. 1998, 70:22-28.
16. Comfort W.W., Raczkowski S.U., Trigos-Arrieta F. Making group topologies with, and without, convergent sequences. Appl. Gen. Topol. 2006, 7(1):109-124.
17. Comfort W.W., Ross K.A. Topologies induced by groups of characters. Fund. Math. 1964, 55:283-291.
18. Dikranjan D. Topologically torsion elements of topological groups. Topology Proc. 2001/2002, 26:505-532.
19. Dikranjan D. Closure operators in topological groups related to von Neumann's kernel. Topology Appl. 2006, 153:1930-1955.
20. Dikranjan D., Di Santo R. A characterization of the circle group via uniqueness of roots. Topology Appl. 2011, 158:159-166.
21. Dikranjan D., Di Santo R. On Armacost's quest on topologically torsion elements. Commun. Algebra 2004, 32:133-146.
22. Dikranjan D., Kunen K. Characterizing subgroups of compact abelian groups. J. Pure Appl. Algebra 2007, 208:285-291.
23. Dikranjan D., Milan C., Tonolo A. A characterization of the maximally almost periodic abelian groups. J. Pure Appl. Algebra 2005, 197(1-3):23-41.
24. Dikranjan D., Prodanov I., Stojanov L. Topological Groups. Characters, Dualities, and Minimal Group Topologies 1990, Marcel Dekker, Inc., New York, Basel.
25. Eggleston H.G. Sets of fractional dimensions which occur in some problems of number theory. Proc. London Math. Soc. (2) 1952, 54:42-93.
26. Gabriyelyan S.S. Groups of quasi-invariance and the Pontryagin duality. Topology Appl. 2010, 157:2786-2802.
27. Gabriyelyan S.S. On T-sequences and characterized subgroups. Topology Appl. 2010, 157:2834-2843.
28. Glicksberg I. Uniform boundedness for groups. Canad. J. Math. 1962, 14:269-276.
29. Graev M. Free topological groups. Izv. Akad. Nauk SSSR Ser. Mat. 1948, 12:278-324.
30. Hart J.E., Kunen K. Limits in function spaces and compact groups. Topology Appl. 2005, 151(1-3):157-168.
31. Hart J.E., Kunen K. Limits in compact Abelian groups. Topology Appl. 2006, 153:991-1002.
32. Hewitt E., Ross K.A. Abstract Harmonic Analysis, vol. I 1979, Springer-Verlag, Berlin. second ed.
33. Kraaikamp C., Liardet P. Good approximations and continued fractions. Proc. Amer. Math. Soc. 1991, 112(2):303-309.
34. Larcher G. A convergence problem connected with continued fractions. Proc. Amer. Math. Soc. 1988, 103(3):718-722.
35. von Neumann J. Almost periodic functions in a group. Trans. Amer. Math. Soc. 1934, 36:445-492.
36. Nienhuys J. Construction of group topologies on Abelian groups. Fund. Math. 1972, 75:101-116.
37. Petersen K., Shapiro L. Induced flows. Trans. Amer. Math. Soc. 1973, 177:375-390.
38. Protasov I.V., Zelenyuk E.G. Topologies on abelian groups. Math. USSR Izv.. Izv. Akad. Nauk SSSR 1990, 54:1090-1107. Russian original:.
39. Protasov I.V., Zelenyuk E.G. Topologies on Groups Determined by Sequences. Math. Stud. Monogr. Ser. 1999, VNTL, Lviv.
40. Raczkowski S.U. Totally bounded topological group topologies on the integers. Topology Appl. 2002, 121:63-74.
41. Vilenkin N. A contribution to the theory of direct decompositions of topological groups. C. R. (Doklady) Acad. Sci. URSS (N.S.) 1945, 47:611-613.
42. Vilenkin N.Ya. The theory of characters of topological Abelian groups with boundedness given. Izv. Akad. Nauk SSSR Ser. Mat. 1951, 15(5):439-462. (in Russian).
43. Winkler R. Ergodic group rotations, Hartman sets and Kronecker sequences. Monatsh. Math. 2002, 135:333-343.