The dual group of a dense subgroup

William Wistar Comfort; S. U. Raczkowski; F. Javier Trigos-Arrieta

Czechoslovak Mathematical Journal (2004)

  • Volume: 54, Issue: 2, page 509-533
  • ISSN: 0011-4642

Abstract

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Throughout this abstract, G is a topological Abelian group and G ^ is the space of continuous homomorphisms from G into the circle group 𝕋 in the compact-open topology. A dense subgroup D of G is said to determine G if the (necessarily continuous) surjective isomorphism G ^ ↠ D ^ given by h ↦ h | D is a homeomorphism, and G is determined if each dense subgroup of G determines G . The principal result in this area, obtained independently by L. Außenhofer and M. J. Chasco, is the following: Every metrizable group is determined. The authors offer several related results, including these. 1. There are (many) nonmetrizable, noncompact, determined groups. 2. If the dense subgroup D i determines G i with G i compact, then ⊕ i D i determines Π i G i . In particular, if each G i is compact then ⊕ i G i determines Π i G i . 3. Let G be a locally bounded group and let G + denote G with its Bohr topology. Then G is determined if and only if G + is determined. 4. Let n o n ( 𝒩 ) be the least cardinal κ such that some X ⊆ 𝕋 of cardinality κ has positive outer measure. No compact G with w ( G ) ≥ n o n ( 𝒩 ) is determined; thus if n o n ( 𝒩 ) = ℵ 1 (in particular if CH holds), an infinite compact group G is determined if and only if w ( G ) = ω . Question. Is there in ZFC a cardinal κ such that a compact group G is determined if and only if w ( G ) < κ ? Is κ = n o n ( 𝒩 ) ? κ = ℵ 1 ?

How to cite

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Comfort, William Wistar, Raczkowski, S. U., and Trigos-Arrieta, F. Javier. "The dual group of a dense subgroup." Czechoslovak Mathematical Journal 54.2 (2004): 509-533. <http://eudml.org/doc/30879>.

@article{Comfort2004,
abstract = {Throughout this abstract, $G$ is a topological Abelian group and $\widehat\{G\}$ is the space of continuous homomorphisms from $G$ into the circle group $\mathbb \{T\}$ in the compact-open topology. A dense subgroup $D$ of $G$ is said to determine $G$ if the (necessarily continuous) surjective isomorphism $\widehat\{G\}\twoheadrightarrow \widehat\{D\}$ given by $h\mapsto h\big |D$ is a homeomorphism, and $G$ is determined if each dense subgroup of $G$ determines $G$. The principal result in this area, obtained independently by L. Außenhofer and M. J. Chasco, is the following: Every metrizable group is determined. The authors offer several related results, including these. 1. There are (many) nonmetrizable, noncompact, determined groups. 2. If the dense subgroup $D_i$ determines $G_i$ with $G_i$ compact, then $\oplus _iD_i$ determines $\Pi _i G_i$. In particular, if each $G_i$ is compact then $\oplus _i G_i$ determines $\Pi _i G_i$. 3. Let $G$ be a locally bounded group and let $G^+$ denote $G$ with its Bohr topology. Then $G$ is determined if and only if $\{G^+\}$ is determined. 4. Let $\mathop \{\mathrm \{n\}on\}(\{\mathcal \{N\}\})$ be the least cardinal $\kappa $ such that some $X \subseteq \{\mathbb \{T\}\}$ of cardinality $\kappa $ has positive outer measure. No compact $G$ with $w(G)\ge \mathop \{\mathrm \{n\}on\}(\{\mathcal \{N\}\})$ is determined; thus if $\mathop \{\mathrm \{n\}on\}(\{\mathcal \{N\}\})=\aleph _1$ (in particular if CH holds), an infinite compact group $G$ is determined if and only if $w(G)=\omega $. Question. Is there in ZFC a cardinal $\kappa $ such that a compact group $G$ is determined if and only if $w(G)<\kappa $? Is $\kappa =\mathop \{\mathrm \{n\}on\}(\{\mathcal \{N\}\})$? $\kappa =\aleph _1$?},
author = {Comfort, William Wistar, Raczkowski, S. U., Trigos-Arrieta, F. Javier},
journal = {Czechoslovak Mathematical Journal},
keywords = {Bohr compactification; Bohr topology; character; character group; Außenhofer-Chasco Theorem; compact-open topology; dense subgroup; determined group; duality; metrizable group; reflexive group; reflective group; Bohr compactification; Bohr topology; character; character group},
language = {eng},
number = {2},
pages = {509-533},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The dual group of a dense subgroup},
url = {http://eudml.org/doc/30879},
volume = {54},
year = {2004},
}

TY - JOUR
AU - Comfort, William Wistar
AU - Raczkowski, S. U.
AU - Trigos-Arrieta, F. Javier
TI - The dual group of a dense subgroup
JO - Czechoslovak Mathematical Journal
PY - 2004
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 54
IS - 2
SP - 509
EP - 533
AB - Throughout this abstract, $G$ is a topological Abelian group and $\widehat{G}$ is the space of continuous homomorphisms from $G$ into the circle group $\mathbb {T}$ in the compact-open topology. A dense subgroup $D$ of $G$ is said to determine $G$ if the (necessarily continuous) surjective isomorphism $\widehat{G}\twoheadrightarrow \widehat{D}$ given by $h\mapsto h\big |D$ is a homeomorphism, and $G$ is determined if each dense subgroup of $G$ determines $G$. The principal result in this area, obtained independently by L. Außenhofer and M. J. Chasco, is the following: Every metrizable group is determined. The authors offer several related results, including these. 1. There are (many) nonmetrizable, noncompact, determined groups. 2. If the dense subgroup $D_i$ determines $G_i$ with $G_i$ compact, then $\oplus _iD_i$ determines $\Pi _i G_i$. In particular, if each $G_i$ is compact then $\oplus _i G_i$ determines $\Pi _i G_i$. 3. Let $G$ be a locally bounded group and let $G^+$ denote $G$ with its Bohr topology. Then $G$ is determined if and only if ${G^+}$ is determined. 4. Let $\mathop {\mathrm {n}on}({\mathcal {N}})$ be the least cardinal $\kappa $ such that some $X \subseteq {\mathbb {T}}$ of cardinality $\kappa $ has positive outer measure. No compact $G$ with $w(G)\ge \mathop {\mathrm {n}on}({\mathcal {N}})$ is determined; thus if $\mathop {\mathrm {n}on}({\mathcal {N}})=\aleph _1$ (in particular if CH holds), an infinite compact group $G$ is determined if and only if $w(G)=\omega $. Question. Is there in ZFC a cardinal $\kappa $ such that a compact group $G$ is determined if and only if $w(G)<\kappa $? Is $\kappa =\mathop {\mathrm {n}on}({\mathcal {N}})$? $\kappa =\aleph _1$?
LA - eng
KW - Bohr compactification; Bohr topology; character; character group; Außenhofer-Chasco Theorem; compact-open topology; dense subgroup; determined group; duality; metrizable group; reflexive group; reflective group; Bohr compactification; Bohr topology; character; character group
UR - http://eudml.org/doc/30879
ER -

References

top
  1. 10.1007/BF01350719, Math. Ann. 177 (1968), 273–277. (1968) MR0232182DOI10.1007/BF01350719
  2. Contributions to the duality theory of Abelian topological groups and to the theory of nuclear groups, PhD. thesis, Universität Tübingen, 1998. (1998) MR1736984
  3. Contributions to the duality theory of Abelian topological groups and to the theory of nuclear groups, Dissertationes Math. Vol. CCCLXXXIV, Warszawa, 1998. (1998) MR1736984
  4. 10.1007/BF01456956, Ann. Math. 264 (1983), 485–493. (1983) Zbl0502.22010MR0716262DOI10.1007/BF01456956
  5. Additive Subgroups of Topological Vector Spaces. Lecture Notes in Mathematics Vol.  1466, Springer-Verlag, Berlin, 1991. (1991) MR1119302
  6. Set Theory: on the Structure of the Real Line, A. K.  Peters, Wellesley, 1990, pp. 546. (1990) MR1350295
  7. 10.2140/pjm.1985.116.217, Pacific J.  Math. 116 (1985), 217–241. (1985) MR0771633DOI10.2140/pjm.1985.116.217
  8. General Topology, Part 2, Addison-Wesley Publishing Company, Reading, Massachusetts, 1966, pp. 363. (1966) Zbl0301.54002MR0205211
  9. 10.1007/s000130050160, Archiv der Math. 70 (1998), 22–28. (1998) Zbl0899.22001MR1487450DOI10.1007/s000130050160
  10. Set Theory for the Working Mathematician, London Mathematical Society Student Texts, Vol. 39 Cambridge University Press, Cambridge, 1997. (1997) Zbl0938.03067MR1475462
  11. 10.1515/form.1994.6.323, Forum Math. 6 (1994), 323–337. (1994) MR1269843DOI10.1515/form.1994.6.323
  12. 10.1007/BF02571718, Math. Z. 215 (1994), 337–346. (1994) MR1262521DOI10.1007/BF02571718
  13. 10.4064/fm-55-3-283-291, Fund. Math. 55 (1964), 283–291. (1964) MR0169940DOI10.4064/fm-55-3-283-291
  14. 10.2140/pjm.1973.49.33, Pacific J.  Math. 49 (1973), 33–44. (1973) MR0372104DOI10.2140/pjm.1973.49.33
  15. The Bohr compactification, modulo a metrizable subgroup, Fund. Math. 143 (1993), 119–136. (1993) MR1240629
  16. 10.1016/0166-8641(91)90096-5, Topology Appl. 41 (1991), 3–15. (1991) MR1129694DOI10.1016/0166-8641(91)90096-5
  17. Topological Groups (Characters, Dualities and Minimal Group Topologies). Monographs and Textbooks in Pure and Applied Mathematics 130, Marcel Dekker, Inc., New York-Basel, 1990. (1990) MR1015288
  18. 10.1016/0166-8641(90)90090-O, Topology Appl. 34 (1990), 69–91. (1990) Zbl0696.22003MR1035461DOI10.1016/0166-8641(90)90090-O
  19. General Topology, Heldermann Verlag, Berlin, 1989. (1989) Zbl0684.54001MR1039321
  20. 10.7146/math.scand.a-10907, Math. Scand. 23 (1968), 169–170. (1968) MR0251457DOI10.7146/math.scand.a-10907
  21. Consequences of Martin’s Axiom. Cambridge Tracts in Mathematics, Vol. 84, Cambridge University Press, Cambridge, 1984. (1984) MR0780933
  22. Infinite Abelian Groups, Vol.  I, Academic Press, New York-San Francisco-London, 1970. (1970) Zbl0209.05503MR0255673
  23. 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. (1996) MR1429651
  24. 10.4153/CJM-1962-017-3, Canad. J.  Math. 14 (1962), 269–276. (1962) Zbl0109.02001MR0155923DOI10.4153/CJM-1962-017-3
  25. Remarks on the cardinality of compact spaces and their Lindelöf subspaces, Proc. Amer. Math. Soc. 59 (1976), 146–148. (1976) MR0423283
  26. Abstract Harmonic Analysis, Vol. I. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Vol.  115, Springer Verlag, Berlin-Göttingen-Heidelberg, 1963. (1963) MR0551496
  27. 10.1007/BF01360918, Math. Ann. 160 (1965), 171–194. (1965) MR0186751DOI10.1007/BF01360918
  28. Real and Abstract Analysis. Graduate Texts in Mathematics Vol. 25, Springer-Verlag, New York, 1965. (1965) 
  29. 10.1007/BFb0069778, Springer-Verlag, Berlin-Heidelberg-New York, 1970. (1970) MR0274648DOI10.1007/BFb0069778
  30. Non-productive duality properties of topological groups, Topology Proc. 25 (2002), 207–216. (2002) MR1925684
  31. A study of topological Abelian groups based on norm space theory, PhD.  thesis, University of Maryland, College Park, 1967. (1967) 
  32. Personal communication, November 20, 2000, . 
  33. Set Theory, Academic Press, Inc., San Diego, 1978. (1978) Zbl0419.03028MR0506523
  34. 10.1215/S0012-7094-48-01557-9, Duke Math.  J. 15 (1948), 649–658. (1948) MR0026999DOI10.1215/S0012-7094-48-01557-9
  35. 10.1215/S0012-7094-50-01737-6, Duke Math.  J. 15 (1950), 419–435. (1950) MR0049906DOI10.1215/S0012-7094-50-01737-6
  36. Classical Descriptive Set Theory. Graduate Texts in Mathematics, Vol.  156, Springer-Verlag, New York, 1994. (1994) MR1321597
  37. 10.1007/BF01361183, Math. Ann. 153 (1964), 150–162. (1964) MR0185417DOI10.1007/BF01361183
  38. Set Theory, An Introduction to Independence Proofs. Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Company, Amsterdam-New York-Oxford, 1980. (1980) MR0597342
  39. 10.1007/BF02988875, Abhandlungen Mathem. Seminar Univ. Hamburg 19 (1955), 244–263. (1955) Zbl0065.01501MR0068545DOI10.1007/BF02988875
  40. Martin’s axiom and topological spaces, Doklady Akad. Nauk SSSR 213 (1973), 532–535. (Russian) (1973) 
  41. k -groups and duality, Trans. Amer. Math. Soc. 151 (1970), 551–561. (1970) Zbl0229.22012MR0270070
  42. Totally bounded groups, PhD. thesis, Wesleyan University, Middletown, 1998. (1998) 
  43. Dense subgroups of locally compact groups, Colloq. Math. 35 (1976), 289–292. (1976) MR0417325
  44. 10.1007/BF01112440, Math. Z. 102 (1967), 225–235. (1967) Zbl0153.04302MR0220000DOI10.1007/BF01112440
  45. Zur Struktur des Verbandes der Gruppentopologien, PhD. thesis, Universität Hannover, Hannover, 1983. (English) (1983) Zbl0547.22004
  46. The number of T 2 -precompact group topologies on free groups, Proc. Amer. Math. Soc. 95 (1985), 315–319. (1985) MR0801346
  47. 10.1090/S0002-9939-1993-1132422-4, Proc. Amer. Math. Soc. 117 (1993), 1195–1200. (1993) MR1132422DOI10.1090/S0002-9939-1993-1132422-4
  48. Locally convex spaces as subgroups of products of locally compact Abelian groups, Math. Japon. 46 (1997), 217–222. (1997) MR1479817
  49. Uniform Structures on Topological Groups and Their Quotients, McGraw-Hill International Book Company, New York-Toronto, 1981. (1981) MR0644485
  50. Topological Vector Spaces, Graduate Texts in Mathematics, Vol. 3, Springer Verlag, New York-Berlin-Heidelberg-Tokyo, 1986, pp. 294. (1986) MR0342978
  51. 10.1512/iumj.1977.26.26079, Indiana Univ. Math.  J. 26 (1977), 981–986. (1977) Zbl0344.46033MR0458134DOI10.1512/iumj.1977.26.26079
  52. 10.2307/1969798, Ann. Math. 56 (1952), 248–253. (1952) MR0049479DOI10.2307/1969798
  53. Additive Gruppen von Folgen ganzer Zahlen, Portugal. Math. 9 (1950), 131–140. (1950) MR0039719
  54. A Course on Borel Sets, Graduate Texts in Mathematics, Vol. 180, Springer-Verlag, New York-Berlin-Heildelberg, 1998. (1998) Zbl0903.28001MR1619545
  55. 10.4064/fm-1-1-93-104, Fund. Math. 1 (1920), 93–104. (1920) DOI10.4064/fm-1-1-93-104
  56. An elementary proof of Steinhaus’ theorem, Proc. Amer. Math. Soc. 36 (1972), 308. (1972) MR0308368
  57. [unknown], Personal Communication, August, 2001. 
  58. 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. (1991) Zbl0777.22003MR1142814
  59. 10.1016/0022-4049(91)90018-W, J.  Pure and Applied Algebra 70 (1991), 199–210. (1991) Zbl0724.22003MR1100517DOI10.1016/0022-4049(91)90018-W
  60. 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. (1990) MR1078647
  61. 10.2140/pjm.1968.26.193, Pacific J.  Math. 26 (1968), 193–196. (1968) Zbl0162.44102MR0232190DOI10.2140/pjm.1968.26.193
  62. Sur les Espaces à Structure Uniforme et sur la Topologie Générale. Publ. Math. Univ. Strasbourg, Vol.  551, (1938), Hermann & Cie, Paris. (1938) 
  63. L’Integration dans les Groupes Topologiques et ses Applications. Actualités Scientifiques et Industrielle, Publ. Math. Inst. Strasbourg, Hermann, Paris, 1951. (1951) 

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