Previous |  Up |  Next


Bohr compactification; Bohr topology; character; character group; Außenhofer-Chasco Theorem; compact-open topology; dense subgroup; determined group; duality; metrizable group; reflexive group; reflective group
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 $\mathop {\mathrm non}({\mathcal N})$ be the least cardinal $\kappa $ such that some $X \subseteq {\mathbb{T}}$ of cardinality $\kappa $ has positive outer measure. No compact $G$ with $w(G)\ge \mathop {\mathrm non}({\mathcal N})$ is determined; thus if $\mathop {\mathrm non}({\mathcal N})=\aleph _1$ (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 $\kappa =\mathop {\mathrm non}({\mathcal N})$? $\kappa =\aleph _1$?
[1] I.  Amemiya and Y. Kōmura: Über nicht-vollständige Montelräme. Math. Ann. 177 (1968), 273–277. DOI 10.1007/BF01350719 | MR 0232182
[2] L. Außenhofer: Contributions to the duality theory of Abelian topological groups and to the theory of nuclear groups. PhD. thesis, Universität Tübingen, 1998. MR 1736984
[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. MR 1736984
[4] W. Banaszczyk: On the existence of exotic Banach-Lie groups. Ann. Math. 264 (1983), 485–493. DOI 10.1007/BF01456956 | MR 0716262 | Zbl 0502.22010
[5] W. Banaszczyk: Additive Subgroups of Topological Vector Spaces. Lecture Notes in Mathematics Vol.  1466. Springer-Verlag, Berlin, 1991. MR 1119302
[6] T. Bartoszyński and H. Judah: Set Theory: on the Structure of the Real Line. A. K.  Peters, Wellesley, 1990, pp. 546. MR 1350295
[7] S. Berhanu, W. W.  Comfort and J. D.  Reid: Counting subgroups and topological group topologies. Pacific J.  Math. 116 (1985), 217–241. DOI 10.2140/pjm.1985.116.217 | MR 0771633
[8] N.  Bourbaki: General Topology, Part 2. Addison-Wesley Publishing Company, Reading, Massachusetts, 1966, pp. 363. MR 0205211 | Zbl 0301.54002
[9] M. J.  Chasco: Pontryagin duality for metrizable groups. Archiv der Math. 70 (1998), 22–28. DOI 10.1007/s000130050160 | MR 1487450 | Zbl 0899.22001
[10] K.  Ciesielski: Set Theory for the Working Mathematician. London Mathematical Society Student Texts, Vol. 39 Cambridge University Press, Cambridge, 1997. MR 1475462 | Zbl 0938.03067
[11] W. W.  Comfort and Dieter Remus: Abelian torsion groups with a pseudocompact group topology. Forum Math. 6 (1994), 323–337. DOI 10.1515/form.1994.6.323 | MR 1269843
[12] W. W.  Comfort and Dieter  Remus: Pseudocompact refinements of compact group topologies. Math. Z. 215 (1994), 337–346. DOI 10.1007/BF02571718 | MR 1262521
[13] W. W.  Comfort and K. A. Ross: Topologies induced by groups of characters. Fund. Math. 55 (1964), 283–291. MR 0169940
[14] W. W.  Comfort and V.  Saks: Countably compact groups and finest totally bounded topologies. Pacific J.  Math. 49 (1973), 33–44. DOI 10.2140/pjm.1973.49.33 | MR 0372104
[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. MR 1240629
[16] W. W.  Comfort and J. van  Mill: Some topological groups with, and some without, proper dense subgroups. Topology Appl. 41 (1991), 3–15. DOI 10.1016/0166-8641(91)90096-5 | MR 1129694
[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. MR 1015288
[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. DOI 10.1016/0166-8641(90)90090-O | MR 1035461 | Zbl 0696.22003
[19] R. Engelking: General Topology. Heldermann Verlag, Berlin, 1989. MR 1039321 | Zbl 0684.54001
[20] P. Flor: Zur Bohr-Konvergenz der Folgen. Math. Scand. 23 (1968), 169–170. MR 0251457
[21] D. H.  Fremlin: Consequences of Martin’s Axiom. Cambridge Tracts in Mathematics, Vol. 84. Cambridge University Press, Cambridge, 1984. MR 0780933
[22] L.  Fuchs: Infinite Abelian Groups, Vol.  I. Academic Press, New York-San Francisco-London, 1970. MR 0255673 | Zbl 0209.05503
[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. MR 1429651
[24] I.  Glicksberg: Uniform boundedness for groups. Canad. J.  Math. 14 (1962), 269–276. DOI 10.4153/CJM-1962-017-3 | MR 0155923 | Zbl 0109.02001
[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. MR 0423283
[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. MR 0551496
[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. DOI 10.1007/BF01360918 | MR 0186751
[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. MR 0274648
[30] M. Higasikawa: Non-productive duality properties of topological groups. Topology Proc. 25 (2002), 207–216. MR 1925684
[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. MR 0506523 | Zbl 0419.03028
[34] S. Kaplan: Extensions of the Pontryagin duality  I: infinite products. Duke Math.  J. 15 (1948), 649–658. DOI 10.1215/S0012-7094-48-01557-9 | MR 0026999
[35] S.  Kaplan: Extensions of the Pontryagin duality II: direct and inverse sequences. Duke Math.  J. 15 (1950), 419–435. DOI 10.1215/S0012-7094-50-01737-6 | MR 0049906
[36] A. S.  Kechris: Classical Descriptive Set Theory. Graduate Texts in Mathematics, Vol.  156. Springer-Verlag, New York, 1994. MR 1321597
[37] Y. Kōmura: Some examples of linear topological spaces. Math. Ann. 153 (1964), 150–162. DOI 10.1007/BF01361183 | MR 0185417
[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. MR 0597342
[39] H. Leptin: Abelsche Gruppen mit kompakten Charaktergruppen und Dualitätstheorie gewisser linear topologischer abelscher Gruppen. Abhandlungen Mathem. Seminar Univ. Hamburg 19 (1955), 244–263. DOI 10.1007/BF02988875 | MR 0068545 | Zbl 0065.01501
[40] V. I.  Malykhin and B. Šapirovskiĭ: Martin’s axiom and topological spaces. Doklady Akad. Nauk SSSR 213 (1973), 532–535. (Russian)
[41] N.  Noble: $k$-groups and duality. Trans. Amer. Math. Soc. 151 (1970), 551–561. MR 0270070 | Zbl 0229.22012
[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. MR 0417325
[44] G. A.  Reid: On sequential convergence in groups. Math. Z. 102 (1967), 225–235. DOI 10.1007/BF01112440 | MR 0220000 | Zbl 0153.04302
[45] D.  Remus: Zur Struktur des Verbandes der Gruppentopologien. PhD. thesis, Universität Hannover, Hannover, 1983. (English) Zbl 0547.22004
[46] D. Remus: The number of $T_2$-precompact group topologies on free groups. Proc. Amer. Math. Soc. 95 (1985), 315–319. MR 0801346
[47] 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. DOI 10.1090/S0002-9939-1993-1132422-4 | MR 1132422
[48] 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. MR 1479817
[49] W. Roelcke and S. Dierolf: Uniform Structures on Topological Groups and Their Quotients. McGraw-Hill International Book Company, New York-Toronto, 1981. MR 0644485
[50] H. H.  Schaefer: Topological Vector Spaces. Graduate Texts in Mathematics, Vol. 3, Springer Verlag, New York-Berlin-Heidelberg-Tokyo, 1986, pp. 294. MR 0342978
[51] S. J.  Sidney: Weakly dense subgroups of Banach spaces. Indiana Univ. Math.  J. 26 (1977), 981–986. DOI 10.1512/iumj.1977.26.26079 | MR 0458134 | Zbl 0344.46033
[52] M. Freundlich Smith: The Pontrjagin duality theorem in linear spaces. Ann. Math. 56 (1952), 248–253. DOI 10.2307/1969798 | MR 0049479
[53] E.  Specker: Additive Gruppen von Folgen ganzer Zahlen. Portugal. Math. 9 (1950), 131–140. MR 0039719
[54] S. M.  Srivastava: A Course on Borel Sets. Graduate Texts in Mathematics, Vol. 180, Springer-Verlag, New York-Berlin-Heildelberg, 1998. MR 1619545 | Zbl 0903.28001
[55] H.  Steinhaus: Sur les distances des points des ensembles de mesure positive. Fund. Math. 1 (1920), 93–104.
[56] K.  Stromberg: An elementary proof of Steinhaus’ theorem. Proc. Amer. Math. Soc. 36 (1972), 308. MR 0308368
[57] S.  Todorčević: Personal Communication, August, 2001.
[58] 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. MR 1142814 | Zbl 0777.22003
[59] F. J.  Trigos-Arrieta: Continuity, boundedness, connectedness and the Lindelöf property for topological groups. J.  Pure and Applied Algebra 70 (1991), 199–210. DOI 10.1016/0022-4049(91)90018-W | MR 1100517 | Zbl 0724.22003
[60] 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. MR 1078647
[61] W. C.  Waterhouse: Dual groups of vector spaces. Pacific J.  Math. 26 (1968), 193–196. DOI 10.2140/pjm.1968.26.193 | MR 0232190 | Zbl 0162.44102
[62] A.  Weil: Sur les Espaces à Structure Uniforme et sur la Topologie Générale. Publ. Math. Univ. Strasbourg, Vol.  551. (1938), Hermann & Cie, Paris.
[63] A. Weil: L’Integration dans les Groupes Topologiques et ses Applications. Actualités Scientifiques et Industrielle, Publ. Math. Inst. Strasbourg. Hermann, Paris, 1951.
Partner of
EuDML logo