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This work contains a number of
theorems about pseudocompact groups. Our first and most useful theorem allows us
to decide whether or not a given (totally bounded) group is pseudocompact on the
basis of how the group sits in its Weil completion. A corollary, which permits us to
answer a question posed by Irving Glicksberg (Trans. Amer. Math. Soc. 90 (1959),
369–382) is: The product of any set of pseudocompact groups is pseudocompact.
Following James Kister (Proc. Amer. Math. Soc. 13 (1962), 37–40) we say that a
topological group G has property U provided that each continuous function mapping
G into the real line is uniformly continuous. We prove that each pseudocompact
group has property U.
Sections 2 and 3 are devoted to solving the following two problems: (a) In order
that a group have property U, is it sufficient that each bounded continuous
real-valued function on it be uniformly continuous? (b) Must a nondiscrete group
with property U be pseudocompact? Theorem 2.8 answers (a) affirmatively. Question
(b), the genesis of this paper, was posed by Kister (loc. cit.). For a large class
of groups the question has an affirmative answer (see 3.1); but in 3.2 we
offer an example (a Lindelöf space) showing that in general the answer is
negative.
Much of the content of this paper is summarized by Theorem 4.1, in which we list
a number of properties equivalent to pseudocompactness for topological
groups. We conclude with an example of a metrizable, non totally bounded
Abelian group on which each uniformly continuous real-valued function is
bounded.
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