\begin{definition}[Definition:Decomposable Set]
A set $S \subseteq \R^n$ is '''decomposable in $m$ sets $A_1, \ldots, A_m \subset \R^n$''' {{iff}} there exist isometries $\phi_1, \ldots, \phi_m: \R^n \to \R^n$ such that:
:$(1):\quad \ds S = \bigcup_{k \mathop = 1}^m \map {\phi_k} {A_k}$ 
:$(2):\quad \forall i \ne j: \map {\phi_i} {A_i} \cap \map {\phi_j} {A_j} = \O$
Such a union is known as a '''decomposition'''.
{{explain|The definition of Definition:Irreducible Space suggests that the definition may apply to a general topological space, not only $\R^n$. This needs to be clarified.}}
\end{definition}