This paper, continuing
previous work by the same author, is concerned with the following problem: Given a
metrisable uniformity U for a set X, does there exist another (distinct)
uniformity B for X such that the two corresponding Hausdorff uniformities
induce the same topology on the set, S(X) say, of all nonempty subsets of
X? Sufficient conditions for the existence, and sufficient conditions for the
nonexistence, of such a uniformity B are given, together with related results
concerning the Hausdorff uniformities (derived from U and B) for S(X1),
where X1 is a subset of X, everywhere dense in the topology derived from
U.
