I have recently given an
example of two different uniformities for the same set X, such that the corresponding
Hausdorff uniformities for the set of nonempty subsets of X are topologically
equivalent; when this is the case we shall call the original uniformities H-equivalent.
The problem posed by Isbell and discussed in a recent paper by D. H. Smith may
therefore be reformulated as follows:- (a) Under what conditions are two
uniformities H-equivalent? (b) Under what conditions does H-equivalence of
uniformities imply identity? The theorems given below supply an answer
to (a) and a partial answer to (b). In particular, they show that neither
Rn nor Qn (Q denoting the set of rational numbers with the usual metric)
has any other uniformity H-equivalent to its metric uniformity. In a sense,
therefore, the example in (1) is the simplest possible one of its kind, though we
give in the course of this paper another simple example using transfinite
ordinals.
