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Some cases of preservation of the Pontryagin dual by taking dense subgroups. (English) Zbl 1235.22004
For an abelian topological group $G$ let $G^\wedge$ denote the group of continuous characters of $G$, endowed with the compact-open topology. A dense subgroup $H$ of an abelian topological group $G$ is said to {\it determine} $G$ if the restriction operator $G^\wedge\to H^\wedge$ is a topological isomorphism. An abelian topological group is {\it determined} if each dense subgroup of $G$ determines $G$. The authors detect some operations preserving determined groups and present several representative examples of determined and non-determined groups. One of the main results is Theorem 14 saying that each compact abelian group $G$ of weight $w(G)\ge \mathfrak c$ contains a dense pseudocompact subgroup which does not determine $G$.

##### MSC:
 22A05 Structure of general topological groups
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##### References:
 [1] S. Ardanza-Trevijano, M. J. Chasco, X. Domínguez, M. Tkachenko, Precompact noncompact reflexive abelian groups, Forum Mathematicum, doi:10.1515/FORM.2011.061, in press. · Zbl 1259.22001 [2] Arkhangelskiı?, A.: Open and close-to-open mappings. Relations among spaces, Trudy moskov. Mat. obsch. 15, 181-223 (1966) [3] Außenhofer, L.: Contributions to the duality theory of abelian topological groups and to the theory of nuclear groups, Diss. math. 384, No. Warszawa (1999) · Zbl 0953.22001 [4] Außenhofer, L.: On the arc component of a locally compact abelian group, Math. Z. 257, No. 2, 239-250 (2007) · Zbl 1136.22003 · doi:10.1007/s00209-007-0108-5 [5] Banaszczyk, W.: Additive subgroups of topological vector spaces, Lecture notes in math. 1466 (1991) · Zbl 0743.46002 [6] Banaszczyk, W.; Chasco, M. J.; Martín-Peinador, E.: Open subgroups and Pontryagin duality, Math. Z. 215, 195-204 (1994) · Zbl 0790.22001 · doi:10.1007/BF02571709 [7] M. Bruguera, M. Tkachenko, Duality in the class of precompact Abelian groups and the Baire property, preprint. · Zbl 1278.43003 [8] Chasco, M. J.: Pontryagin duality for metrizable groups, Arch. math. 70, 22-28 (1998) · Zbl 0899.22001 · doi:10.1007/s000130050160 [9] Comfort, W. W.; Raczkowski, S. U.; Trigos-Arrieta, F. J.: The dual group of a dense subgroup, Czech math. J. 54, 509-533 (2004) · Zbl 1080.22500 · doi:10.1023/B:CMAJ.0000042588.07352.99 [10] Dikranjan, D.; Shakhmatov, D.: Quasi-convex density and determining subgroups of compact abelian groups, J. math. Anal. appl. 363, 42-48 (2010) · Zbl 1178.22005 · doi:10.1016/j.jmaa.2009.07.038 [11] D. Dikranjan, D. Shakhmatov, Which subgroups determine a compact abelian group?, preprint. · Zbl 1178.22005 [12] Dikranjan, D.; Tkachenko, M.: Sequential completeness of quotient groups, Bull. austral. Math. soc. 61, 129-150 (2000) · Zbl 0943.22001 · doi:10.1017/S0004972700022085 [13] Ferrer, M. V.; Hernández, S.: Dual topologies of non-abelian groups · Zbl 1246.22008 [14] Fuchs, L.: Infinite abelian groups, vol. I, Pure and applied mathematics 36 (1970) · Zbl 0209.05503 [15] Galindo, J.; Macario, S.: Pseudocompact group topologies with no infinite compact subsets, J. pure appl. Algebra 215, No. 4, 655-663 (2011) · Zbl 1215.54015 · doi:10.1016/j.jpaa.2010.06.014 [16] Hernández, S.; Macario, S.; Trigos-Arrieta, F. J.: Uncountable products of determined groups need not be determined, J. math. Anal. appl. 348, No. 2, 834-842 (2008) · Zbl 1156.22002 · doi:10.1016/j.jmaa.2008.07.065 [17] Hewitt, E.; Ross, K. A.: Abstract harmonic analysis I, (1979) · Zbl 0416.43001 [18] Kaplan, S.: Extensions of the Pontryagin duality I: Infinite products, Duke math. J. 15, 649-658 (1948) · Zbl 0034.30601 · doi:10.1215/S0012-7094-48-01557-9 [19] S.U. Raczkowski-Trigos, Totally bounded groups, Ph.D. thesis, Wesleyan University, Middletown, 1998. [20] Rajagopalan, M.; Subrahmanian, H.: Dense subgroups of locally compact groups, Colloq. math. 35, 289-292 (1976) · Zbl 0331.22005 [21] Willard, S.: General topology, (1970) · Zbl 0205.26601