A metric space X has the unique midset property if there is
a topology-preserving metric d on X such
that for every pair of distinct points x, y there is one and only one point
p such that d(x, p) = d(y, p). The
following are proved. (1) The discrete space with cardinality [nfr ] has the unique midset property if and only
if [nfr ] ≠ 2, 4 and [nfr ] [les ] [cfr ], where [cfr ] is the cardinality of the
continuum. (2) If D is a discrete space with cardinality
not greater than [cfr ], then the countable power DN
of D has the unique midset property. In particular, the
Cantor set and the space of irrational numbers have the unique midset property.
A finite discrete space with n points has the unique midset property if and only if there is an edge
colouring ϕ of the complete graph Kn such that for
every pair of distinct vertices x, y there is one and only
one vertex p such that ϕ(xp) = ϕ(yp). Let
ump(Kn) be the smallest number of colours necessary for such
a colouring of Kn. The following are proved. (3) For each
k [ges ] 0, ump(K2k+1) = k. (4) For each k [ges ] 3,
k [les ] ump(K2k) [les ] 2k−1.