A metric space $\text{M=}\left( M;\text{d} \right)$ is homogeneous if for every isometry $f$ of a finite subspace of $\text{M}$ to a subspace of $\text{M}$ there exists an isometry of $\text{M}$ onto $\text{M}$ extending $f$. The space $\text{M}$ is universal if it isometrically embeds every finite metric space $\text{F}$ with $\text{dist}\left( \text{F} \right)\subseteq \text{dist}\left( \text{M} \right)$ (with $\text{dist}\left( \text{M} \right)$ being the set of distances between points in $\text{M}$).

A metric space $U$ is a Urysohn metric space if it is homogeneous, universal, separable, and complete. (We deduce as a corollary that a Urysohn metric space $U$ isometrically embeds every separable metric space $\text{M}$ with $\text{dist}\left( \text{M} \right)\subseteq \text{dist}\left( U \right)$.)

The main results are: (1) A characterization of the sets $\text{dist}\left( U \right)$ for Urysohn metric spaces $U$. (2) If $R$ is the distance set of a Urysohn metric space and $\text{M}$ and $\text{N}$ are two metric spaces, of any cardinality with distances in $R$, then they amalgamate disjointly to a metric space with distances in $R$. (3) The completion of every homogeneous, universal, separable metric space $\text{M}$ is homogeneous.

We investigate extensions of S. Solecki's theorem on closing off finite partial isometries of metric spaces [11] and obtain the following exact equivalence: any action of a discrete group Γ by isometries of a metric space is finitely approximable if and only if any product of finitely generated subgroups of Γ is closed in the profinite topology on Γ.

Given a finitely generated group Γ we study the space Isom(Γ, ) of all actions of Γ by isometries of the rational Urysohn metric space , where Isom (Γ, ) is equipped with the topology it inherits seen as a closed subset of Isom . When Γ is the free group on n generators this space is just Isom , but is in general significantly more complicated. We prove that when Γ is finitely generated Abelian there is a generic point in Isom(Γ, ), i.e., there is a comeagre set of mutually conjugate isometric actions of Γ on .