Topology Proceedings 28 No. 2 (2004), pp. 569-577

Functional balance, discrete balance, and balance in topological groups

Gerald Itzkowitz

We consider the question of determining parameters for when topological group balance (the left and right uniformities on the group are equivalent) and functional balance (the classes of left and right uniformly continuous bounded real valued functions coincide) in topological groups are equivalent. Our main result is that a topological group G is balanced iff it is functionally balanced and discretely balanced. A topological group is discretely balanced if every left uniformly discrete subset is right uniformly discrete. This partially answers a question of T.S. Wu. The proof makes use of a theorem derived from the well known theorem of Katetov on extending real valued bounded uniformly continuous functions from a subspace of a uniform space to the whole space and a characterization of uniform separation pointed out by the author in a previous paper. It is still unknown if the conditions that G is balanced and G is functionally balanced are equivalent.

Keywords: T₀ topological group, uniform space, uniformly continuous function, uniform separation, functionally uniformly separated, left uniformity, right uniformity, balanced group, left uniformly continuous function, right uniformly continuous function, functionally balanced group, left uniformly discrete set, right uniformly discrete set, discretely balanced group, strongly uniformly discrete

Mathematics Subject Classification: 22C05 22A05 54A25 54B05 54B10 54B15

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Topology Proceedings, Volume 28, No. 2 (2004)
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