  Prompted by a recent question of G. Hjorth as to whether a bounded Urysohn
space is indivisible, that is to say has the property that any partition into
finitely many pieces has one piece which contains an isometric copy of the
space, we answer this question and more generally investigate partitions of
countable metric spaces. We show that an indivisible metric space must be
totally Cantor disconnected, which implies in particular that every Urysohn
space U_V with V bounded or not but dense in some initial segment of R+, is
divisible. On the other hand we also show that one can remove "large" pieces
from a bounded Urysohn space with the remainder still inducing a copy of this
space, providing a certain "measure" of the indivisibility. Associated with
every totally Cantor disconnected space is an ultrametric space, and we go on
to characterize the countable ultrametric spaces which are homogeneous and
indivisible.
