The dual group of a dense subgroup

Czechoslovak Mathematical Journal

Volumn 54, Issue 2, 2004, Pages 509-533

  Comfort, W W ^a   Raczkowski, S U ^b   Trigos Arrieta, F Javier ^b  

^a WESLEYAN UNIVERSITY   (United States)

^b CALIFORNIA STATE UNIVERSITY   (United States)

Author keywords

Au enhofer Chasco Theorem; Bohr compactification; Bohr topology; character; character group; compact open topology; dense subgroup; determined group; duality; metrizable group; reflective group; reflexive group

Indexed keywords

EID: 4944255084     PISSN: 00114642     EISSN: None     Source Type: Journal    
DOI: 10.1023/B:CMAJ.0000042588.07352.99     Document Type: Article

References (64)

1. I. Amemiya and Y. Kōmura: Über nicht-vollständige Montelräme. Math. Ann. 177 (1968), 273-277.
2. L. Außenhofer: Contributions to the duality theory of Abelian topological groups and to the theory of nuclear groups. PhD. thesis. Universität Tübingen, 1998.
3. L. Außenhofer: Contributions to the duality theory of Abelian topological groups and to the theory of nuclear groups. Dissertationes Math. Vol. CCCLXXXIV. Warszawa, 1998.
4. W. Banaszczyk: On the existence of exotic Banach-Lie groups. Ann. Math. 264 (1983), 485-493.
5. W. Banaszczyk: Additive Subgroups of Topological Vector Spaces. Lecture Notes in Mathematics Vol. 1466. Springer-Verlag, Berlin, 1991.
6. T. Bartoszyński and H. Judah: Set Theory: on the Structure of the Real Line. A. K. Peters, Wellesley, 1990, pp. 546.
7. S. Berhanu, W. W. Comfort and J. D. Reid: Counting subgroups and topological group topologies. Pacific J. Math. 116 (1985), 217-241.
8. N. Bourbaki: General Topology, Part 2. Addison-Wesley Publishing Company, Reading, Massachusetts, 1966, pp. 363.
9. M. J. Chasco: Pontryagin duality for metrizable groups. Archiv der Math. 70 (1998), 22-28.
10. K. Ciesielski: Set Theory for the Working Mathematician. Cambridge University Press, Cambridge, 1997.
11. W. W. Comfort and Dieter Remus: Abelian torsion groups with a pseudocompact group topology. Forum Math. 6 (1994), 323-337.
12. W. W. Comfort and Dieter Remus: Pseudocompact refinements of compact group topologies. Math. Z. 215 (1994), 337-346.
13. W. W. Comfort and K. A. Ross: Topologies induced by groups of characters. Fund. Math. 55 (1964), 283-291.
14. W. W. Comfort and V. Saks: Countably compact groups and finest totally bounded topologies. Pacific J. Math. 49 (1973), 33-44.
15. W. W. Comfort, F. Javier Trigos-Arrieta and and Ta-Sun Wu: The Bohr compactification, modulo a metrizable subgroup. Fund. Math. 143 (1993), 119-136; Correction: 152 (1997), 97-98.
16. W. W. Comfort and J. van Mill: Some topological groups with, and some without, proper dense subgroups. Topology Appl. 41 (1991), 3-15.
17. D. N. Dikranjan, I. R. Prodanov and L. N. Stoyanov: Topological Groups (Characters, Dualities and Minimal Group Topologies). Monographs and Textbooks in Pure and Applied Mathematics 130. Marcel Dekker, Inc., New York-Basel, 1990.
18. E. K. van Douwen: The maximal totally bounded group topology on G and the biggest minimal G-space for Abelian groups G. Topology Appl. 34 (1990), 69-91.
19. R. Engelking: General Topology. Heldermann Verlag, Berlin, 1989.
20. P. Flor: Zur Bohr-Konvergenz der Folgen. Math. Scand. 23 (1968), 169-170.
21. D. H. Fremlin: Consequences of Martin's Axiom. Cambridge Tracts in Mathematics, Vol. 84. Cambridge University Press, Cambridge, 1984.
22. L. Fuchs: Infinite Abelian Groups, Vol. I. Academic Press, New York-San Francisco-London, 1970.
23. J. Galindo and S. Hernández: On the completion of a MAP group. In: Papers on General Topology and Applications. Proc. Eleventh (August, 1995) summer topology conference at the University of Maine. Annals New York Acad. Sci. Vol. 806 (S. Andima, R. C. Flagg, G. Itzkowitz, Yung Kong, R. Kopperman, and P. Misra, eds.). New York, 1996, pp. 164-168.
24. I. Glicksberg: Uniform boundedness for groups. Canad. J. Math. 14 (1962), 269-276.
25. A. Hajnal and I. Juhász: Remarks on the cardinality of compact spaces and their Lindelöf subspaces. Proc. Amer. Math. Soc. 59 (1976), 146-148.
26. E. Hewitt and K. A. Ross: Abstract Harmonic Analysis, Vol. I. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Vol. 115. Springer Verlag, Berlin-Göttingen-Heidelberg, 1963.
27. E. Hewitt and K. A. Ross: Extensions of Haar measure and of harmonic analysis for locally compact Abelian groups. Math. Ann. 160 (1965), 171-194.
28. E. Hewitt and K. R. Stromberg: Real and Abstract Analysis. Graduate Texts in Mathematics Vol. 25. Springer-Verlag, New York, 1965.
29. H. Heyer: Dualität lokalkompakter Gruppen. Lecture Notes in Mathematics Vol. 150. Springer-Verlag, Berlin-Heidelberg-New York, 1970.
30. M. Higasikawa: Non-productive duality properties of topological groups. Topology Proc. 25 (2002), 207-216.
31. R. Hooper: A study of topological Abelian groups based on norm space theory. PhD. thesis. University of Maryland, College Park, 1967.
32. M. Hrušák: Personal communication, November 20, 2000.
33. T. Jech: Set Theory. Academic Press, Inc., San Diego, 1978.
34. S. Kaplan: Extensions of the Pontryagin duality I: infinite products. Duke Math. J. 15 (1948), 649-658.
35. S. Kaplan: Extensions of the Pontryagin duality II: direct and inverse sequences. Duke Math. J. 15 (1950), 419-435.
36. A. S. Kechris: Classical Descriptive Set Theory. Graduate Texts in Mathematics, Vol. 156. Springer-Verlag, New York, 1994.
37. Y. Kōmura: Some examples of linear topological spaces. Math. Ann. 153 (1964), 150-162.
38. K. Kunen: Set Theory, An Introduction to Independence Proofs. Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Company, Amsterdam-New York-Oxford, 1980.
39. H. Leptin: Abelsche Gruppen mit kompakten Charaktergruppen und Dualitätstheorie gewisser linear topologischer abelscher Gruppen. Abhandlungen Mathem. Seminar Univ. Hamburg 19 (1955), 244-263.
40. V. I. Malykhin and B. Šapirovskiǐ: Martin's axiom and topological spaces. Doklady Akad. Nauk SSSR 213 (1973), 532-535. (In Russian.)
41. N. Noble: k-groups and duality. Trans. Amer. Math. Soc. 151 (1970), 551-561.
42. S. U. Raczkowski-Trigos: Totally bounded groups. PhD. thesis. Wesleyan University, Middletown, 1998.
43. M. Rajagopalan and H. Subrahmanian: Dense subgroups of locally compact groups. Colloq. Math. 35 (1976), 289-292.
44. G. A. Reid: On sequential convergence in groups. Math. Z. 102 (1967), 225-235.
45. D. Remus: Zur Struktur des Verbandes der Gruppentopologien. PhD. thesis. Universität Hannover, Hannover, 1983 (In English.); Summary: Resultate Math. 6 (1983), 151-152.
46. D. Remus: Zur Struktur des Verbandes der Gruppentopologien. PhD. thesis. Universität Hannover, Hannover, 1983 (In English.); Summary: Resultate Math. 6 (1983), 151-152.
47. 2-precompact group topologies on free groups. Proc. Amer. Math. Soc. 95 (1985), 315-319.
48. D. Remus and F. Javier Trigos-Arrieta: Abelian groups which satisfy Pontryagin duality need not respect compactness. Proc. Amer. Math. Soc. 117 (1993), 1195-1200.
49. D. Remus and F. Javier Trigos-Arrieta: Locally convex spaces as subgroups of products of locally compact Abelian groups. Math. Japon. 46 (1997), 217-222.
50. W. Roelcke and S. Dierolf: Uniform Structures on Topological Groups and Their Quotients. McGraw-Hill International Book Company, New York-Toronto, 1981.
51. H. H. Schaefer: Topological Vector Spaces. Graduate Texts in Mathematics, Vol. 3. Springer Verlag, New York-Berlin-Heidelberg-Tokyo, 1986, pp. 294.
52. S. J. Sidney: Weakly dense subgroups of Banach spaces. Indiana Univ. Math. J. 26 (1977), 981-986.
53. M. Freundlich Smith: The Pontrjagin duality theorem in linear spaces. Ann. Math. 56 (1952), 248-253.
54. E. Specker: Additive Gruppen von Folgen ganzer Zahlen. Portugal. Math. 9 (1950), 131-140.
55. S. M. Srivastava: A Course on Borel Sets. Graduate Texts in Mathematics, Vol. 180. Springer-Verlag, New York-Berlin-Heildelberg, 1998.
56. H. Steinhaus: Sur les distances des points des ensembles de mesure positive. Fund. Math. 1 (1920), 93-104 47.0179.02.
57. K. Stromberg: An elementary proof of Steinhaus' theorem. Proc. Amer. Math. Soc. 36 (1972), 308.
58. S. Todorčević:. Personal Communication, August, 2001.
59. F. J. Trigos-Arrieta: Pseudocompactness on groups. In: General Topology and Applications (S. J. Andima, R. Kopperman, P. R. Misra, J. Z. Reichman, and A. R. Todd, eds.). Marcel Dekker, Inc., New York-Basel-Hong Kong, 1991, pp. 369-378.
60. F. J. Trigos-Arrieta: Continuity, boundedness, connectedness and the Lindelöf property for topological groups. J. Pure and Applied Algebra 70 (1991), 199-210.
61. J. E. Vaughan: Small uncountable cardinals and topology. In: Open Problems in Topology, Chapter 11 (J. van Mill, G. M. Reed, eds.). Elsevier Science Publishers (B. V.), Amsterdam-New York-Oxford-Tokyo, 1990.
62. W. C. Waterhouse: Dual groups of vector spaces. Pacific J. Math. 26 (1968), 193-196.
63. A. Weil: Sur les Espaces à Structure Uniforme et sur la Topologie Générale. Publ. Math. Univ. Strasbourg, Vol. 551. Hermann & Cie, Paris.
64. A. Weil: L'Integration dans les Groupes Topologiques et ses Applications. Actualités Scientifiques et Industrielle, Publ. Math. Inst. Strasbourg. Hermann, Paris, 1951.