zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Some cases of preservation of the Pontryagin dual by taking dense subgroups. (English) Zbl 1235.22004
For an abelian topological group G let G 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 determine G if the restriction operator G H is a topological isomorphism. An abelian topological group is 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)𝔠 contains a dense pseudocompact subgroup which does not determine G.
MSC:
22A05Structure of general topological groups
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.
[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.
[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.
[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
[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)
[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