Definition:Decomposable Set

Definition

A set $S \subseteq \R^n$ is decomposable in $m$ sets $A_1, \ldots, A_m \subset \R^n$ if and only if 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.

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Also see

• Decomposition of Set is Partition

• Definition:Irreducible Space: a set which can not be decomposed