Zbl 1007.22005
Comfort, W.W.;
Raczkowski, S.U.;
TrigosArrieta, F.Javier
Concerning the dual group of a dense subgroup. (English)
[CA] Simon, Petr (ed.), Proceedings of the 9th Prague topological symposium,
Prague, Czech Republic, August 1925, 2001. Toronto: Topology Atlas. 2335,
electronic only (2002). [ISBN 097308670X]
A topological Abelian group $G$ is called {\it determined} if every dense
subgroup of $G$ determines $G$, meaning that its dual group coincides
(topologically) with that of $G$. This concept was studied previously by {\it
L. Aussenhofer} [Diss. Math. 384, 113 p. (1999;
Zbl 0953.22001)] and by {\it M. J. Chasco} [Arch. Math. 70, 2228 (1998;
Zbl 0899.22001)]. These authors had shown that every metrizable group is
determined. The present paper, an expanded version of the second author's Ph.D.
thesis, states many more results on determined groups. Thus, assuming the
continuum hypothesis, a compact group is determined if and only if its weight
is $\omega$. Whether this holds without the continuum hypothesis is an open
question. A large class of examples of determined groups is constructed in the
following way: let $G$ be locally bounded Abelian and determined; on $G$,
replace the given topology by the Bohr topology. The resulting group $H$ will
be determined. If $G$ is not totally bounded then $H$ is totally bounded but
not metrizable.  A paper containing full proofs is announced.
[
Peter Flor (Graz) ]
