# Definition:Decomposable Set

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## Definition

A set $S \subset \R^n$ is **decomposable in $m$ sets $A_1, \ldots, A_m \subset \R^n$** if there exist isometries $\phi_1, \ldots, \phi_m: \R^n \to \R^n$ such that:

- $(1):\quad \displaystyle S = \bigcup_{k \mathop = 1}^m \phi_k \left({A_k}\right)$
- $(2):\quad \forall i \ne j: \phi_i \left({A_i}\right) \cap \phi_j \left({A_j}\right) = \varnothing$

Such a union is known as a **decomposition**.

## Also see

- Definition:Irreducible Space: a set which can not be
**decomposed**