Prompted by a recent question of G. Hjorth [12] 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.
It was proved few years ago that classes of Boolean functions definable by means of functional equations [9], or equivalently, by means of relational constraints [15], coincide with initial segments of the quasi-ordered set (Ω, ≤) made of the set Ω of Boolean functions, suitably quasi-ordered. The resulting ordered set (Ω/ ≡, ⊑) embeds into ([ω] <ω , ⊆), the set -ordered by inclusion-of finite subsets of the set ω of integers. We prove that (Ω/ ≡, ⊑) also embeds ([ω] <ω , ⊆). We prove that initial segments of (Ω, ≤) which are definable by finitely many obstructions coincide with classes defined by finitely many equations. This gives, in particular, that the classes of Boolean functions with a bounded number of essential variables are finitely definable. As an example, we provide a concrete characterization of the subclasses made of linear functions.
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