I asserted in (1), p. 35, that the topology of the hyperspace H(μX)of a uniform space μX determines the uniformity μ, and sketched a proof. D. Hammond Smith has shown (2) that the proof indicated is wrong, even for metric spaces. However, Smith proved the assertion for metric spaces and some others, and left open the question whether it was true in general. This note gives a counter example. I should note that I began with a transfinite construction based on Smith's example; whether that would have worked or not, the following construction is simpler.