HewittMarczewskiPondiczery type theorem for abelian groups and Markov's potential density
Authors:
Dikran Dikranjan and Dmitri Shakhmatov
Journal:
Proc. Amer. Math. Soc. 138 (2010), 29792990
MSC (2010):
Primary 22A05; Secondary 20K99, 22C05, 54A25, 54B10, 54D65
Published electronically:
April 1, 2010
MathSciNet review:
2644909
Fulltext PDF
Abstract 
References 
Similar Articles 
Additional Information
Abstract: For an uncountable cardinal and a subset of an abelian group , the following conditions are equivalent:  (i)
 for all integers ;
 (ii)
 there exists a group homomorphism such that is dense in .
Moreover, if , then the following item can be added to this list:  (iii)
 there exists an isomorphism between and a subgroup of such that is dense in .
We prove that the following conditions are equivalent for an uncountable subset of an abelian group that is either (almost) torsionfree or divisible:  (a)
 is dense in for some Hausdorff group topology on ;
 (b)
 is dense in some precompact Hausdorff group topology on ;
 (c)
 for every integer .
This partially resolves a question of Markov going back to 1946.
 1.
Dikran
N. Dikranjan, Ivan
R. Prodanov, and Luchezar
N. Stoyanov, Topological groups, Monographs and Textbooks in
Pure and Applied Mathematics, vol. 130, Marcel Dekker, Inc., New York,
1990. Characters, dualities and minimal group topologies. MR 1015288
(91e:22001)
 2.
Dikran
Dikranjan and Dmitri
Shakhmatov, Algebraic structure of pseudocompact groups, Mem.
Amer. Math. Soc. 133 (1998), no. 633, x+83. MR 1396956
(98j:22001), 10.1090/memo/0633
 3.
Dikran
Dikranjan and Dmitri
Shakhmatov, Forcing hereditarily separable compactlike group
topologies on abelian groups, Topology Appl. 151
(2005), no. 13, 2–54. MR 2139740
(2006d:22002), 10.1016/j.topol.2004.07.012
 4.
D. Dikranjan and D. Shakhmatov, Selected topics from the structure theory of topological groups, pp. 389406 in: Open Problems in Topology. II (E. Perl, ed.), Elsevier, 2007.
 5.
D. Dikranjan and D. Shakhmatov, The MarkovZariski topology of an abelian group, J. Algebra, to appear.
 6.
D. Dikranjan and D. Shakhmatov, A KroneckerWeyl theorem for subsets of Abelian groups, submitted.
 7.
Dikran
Dikranjan and Michael
Tkačenko, Weakly complete free topological groups,
Topology Appl. 112 (2001), no. 3, 259–287. MR 1824163
(2002b:22004), 10.1016/S01668641(99)002357
 8.
Ryszard
Engelking, General topology, 2nd ed., Sigma Series in Pure
Mathematics, vol. 6, Heldermann Verlag, Berlin, 1989. Translated from
the Polish by the author. MR 1039321
(91c:54001)
 9.
A.
Markoff, On unconditionally closed sets, Rec. Math. [Mat.
Sbornik] N.S. 18(60) (1946), 3–28 (Russian., with
English summary). MR 0015395
(7,412b)
 10.
Mikhail
Tkachenko and Ivan
Yaschenko, Independent group topologies on abelian groups,
Proceedings of the International Conference on Topology and its
Applications (Yokohama, 1999), 2002, pp. 425–451. MR 1919318
(2003h:22003), 10.1016/S01668641(01)001614
 11.
Hermann
Weyl, Über die Gleichverteilung von Zahlen mod. Eins,
Math. Ann. 77 (1916), no. 3, 313–352 (German).
MR
1511862, 10.1007/BF01475864
 1.
 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 YorkBasel, 1990. MR 1015288 (91e:22001)
 2.
 D. Dikranjan and D. Shakhmatov, Algebraic structure of pseudocompact groups, Mem. Amer. Math. Soc. 133 (1998), 83 pages. MR 1396956 (98j:22001)
 3.
 D. Dikranjan and D. Shakhmatov, Forcing hereditarily separable compactlike group topologies on abelian groups, Topology Appl. 151 (2005), 254. MR 2139740 (2006d:22002)
 4.
 D. Dikranjan and D. Shakhmatov, Selected topics from the structure theory of topological groups, pp. 389406 in: Open Problems in Topology. II (E. Perl, ed.), Elsevier, 2007.
 5.
 D. Dikranjan and D. Shakhmatov, The MarkovZariski topology of an abelian group, J. Algebra, to appear.
 6.
 D. Dikranjan and D. Shakhmatov, A KroneckerWeyl theorem for subsets of Abelian groups, submitted.
 7.
 D. Dikranjan and M. Tkačenko, Weakly complete free topological groups, Topology Appl. 112 (2001), 259287. MR 1824163 (2002b:22004)
 8.
 R. Engel'king, General topology (second edition), Sigma Series in Pure Mathematics, 6, Heldermann Verlag, Berlin, 1989. MR 1039321 (91c:54001)
 9.
 A. A. Markov, On unconditionally closed sets (Russian), Mat. Sbornik 18 (1946), 328; English translation in: A. A. Markov, Three papers on topological groups: I. On the existence of periodic connected topological groups, II. On free topological groups, III. On unconditionally closed sets, Amer. Math. Soc. Translation 1950 (1950), no. 30, 120 pp.; another English translation in: Topology and Topological Algebra, Translations Series 1, vol. 8, pp. 273304, Amer. Math. Soc., 1962. MR 0015395 (7:412b) MR0037854 (12:318b)
 10.
 M. Tkachenko and I. Yaschenko, Independent group topologies on abelian groups, Topology Appl. 122 (2002), 425451. MR 1919318 (2003h:22003)
 11.
 H. Weyl, Über die Gleichverteilung von Zahlen mod. Eins (German), Math. Ann. 77 (1916), no. 3, 313352. MR 1511862
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2010):
22A05,
20K99,
22C05,
54A25,
54B10,
54D65
Retrieve articles in all journals
with MSC (2010):
22A05,
20K99,
22C05,
54A25,
54B10,
54D65
Additional Information
Dikran Dikranjan
Affiliation:
Università di Udine, Dipartimento di Matematica e Informatica, via delle Scienze, 206  33100 Udine, Italy
Email:
dikran.dikranjan@dimi.uniud.it
Dmitri Shakhmatov
Affiliation:
Division of Mathematics, Physics and Earth Sciences, Graduate School of Science and Engineering, Ehime University, Matsuyama 7908577, Japan
Email:
dmitri@dpc.ehimeu.ac.jp
DOI:
http://dx.doi.org/10.1090/S0002993910103025
Keywords:
Abelian group,
monomorphism,
homomorphism,
potentially dense set,
dense subset,
precompact group
Received by editor(s):
November 10, 2008
Received by editor(s) in revised form:
November 24, 2009
Published electronically:
April 1, 2010
Additional Notes:
The first author was partially supported by SRA, grants P102920101 and J196430101 and by grant MTM200914409C0201
The second author was partially supported by GrantinAid for Scientific Research (C) No. 19540092 of the Japan Society for the Promotion of Science (JSPS)
Communicated by:
Alexander N. Dranishnikov
Article copyright:
© Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
