Vol. 22, No. 1, 1967

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On H-equivalence of uniformities: The Isbell-Smith problem

A. J. Ward

Vol. 22 (1967), No. 1, 189–196

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.

Mathematical Subject Classification
Primary: 54.30
Received: 24 October 1965
Published: 1 July 1967
A. J. Ward