Topology Atlas
Document # ppae04
Concerning the dual group of a dense subgroup
W. W. Comfort, S. U. Raczkowski and F. Javier TrigosArrieta
Proceedings of the Ninth Prague Topological Symposium
(2001)
pp. 2335
Throughout this Abstract, G is a topological Abelian group and
[^G] is the space of continuous homomorphisms from G into
T in the compactopen topology. A dense subgroup D of G
determines G if the (necessarily continuous)
surjective isomorphism [^G]\twoheadrightarrow[^D] given by
h > hD 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. Auß enhofer and
M. J. Chasco, is the following: Every metrizable group is
determined.
The authors offer several related results, including these.
 There are (many) nonmetrizable, noncompact, determined groups.

If the dense subgroup D_{i} determines G_{i} with G_{i} compact,
then \oplus_{i} D_{i} determines \Pi_{i} G_{i}. In particular, if each
G_{i} is compact then \oplus_{i} G_{i} determines \Pi_{i} G_{i}.

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.

Let \non(N) be the least cardinal \kappa such that some
X subset or equal \TT of cardinality \kappa has positive outer measure.
No compact G with w(G) >= \non(N) is determined; thus if
\non(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 = \non(N)?
\kappa = \aleph_{1}?
Mathematics Subject Classification. 22A10 22B99 22C05 43A40 54H11 (03E35 03E50 54D30 54E35).
Keywords. Bohr compactification, Bohr topology, character, character
group, Au{\ss}enhoferChasco Theorem, compactopen topology, dense
subgroup, determined group, duality, metrizable group, reflexive group,
reflective group.
 Document formats
 AtlasImage (for online previewing)
 LaTeX 40.2 Kb
 DVI 57.4 Kb
 PostScript 227.7 Kb
 gzipped PostScript 91.1 Kb
 PDF 269.1 Kb
 arXiv
 math.GN/0204147
 Metadata
 Citation
 Reference list in BibTeX
Comments. A full version of this article, with complete proofs, will be submitted for publication elsewhere.
Copyright © 2002
Charles University and
Topology Atlas.
Published April 2002.