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Pseudocompactness on groups. (English) Zbl 0777.22003
General topology and applications, Proc. 5th Northeast Conf., New York/NY (USA) 1989, Lect. Notes Pure Appl. Math. 134, 369-378 (1991).
[For the entire collection see Zbl 0744.00027.]
Let $$G$$ be a locally compact Abelian group with character group $$G^ \wedge$$. The author denotes by $$G^ +$$ the underlying group $$G$$ endowed with the weakest topology making every character from $$G^ \wedge$$ continuous. It is a theorem of I. Glicksberg [Can. J. Math. 14, 269-276 (1962; Zbl 0109.020)] that $$G$$ and $$G^ +$$ have the same compact subsets. In this paper, a new proof of Glicksberg’s theorem is given without using tools from functional analysis. The case if $$G$$ is discrete was treated by W. W. Comfort and the author [Lect. Notes Pure Appl. Math. 134, 25-33 (1991; see the preceding review Zbl 0777.22002)]. It is a key result for the proof of the general theorem given in this paper. Finally the author shows that $$G$$ and $$G^ +$$ have the same pseudocompact subsets.
{Reviewer’s remark: Further results concerning relationships between $$G$$ and $$G^ +$$ can be found in the author’s paper [J. Pure Appl. Algebra 70, No. 1/2, 199-210 (1991; Zbl 0724.22003)]}.
Reviewer: D.Remus (Hagen)

##### MSC:
 22B05 General properties and structure of LCA groups 54A05 Topological spaces and generalizations (closure spaces, etc.) 20K45 Topological methods for abelian groups 43A40 Character groups and dual objects