Decomposition

A decomposition is a representation of a given set as the union of a system of pairwise disjoint sets.

In discrete geometry one frequently considers the decomposition of some space into closed domains which cover the entire space and whose pairwise intersections have no interior points (they may have common boundary points). For example, if one fixes any point lattice in the Euclidean space $\mathbf R^n$ and assigns to each point of the lattice those points of the space that are less distant from this point than from any other point of the lattice, then the so-called Dirichlet–Voronoi decomposition is obtained. A decomposition of a space $P$ is called regular if for any domains $D_1$ and $D_2$ in it there is a motion $M$ for which $D_2=M(D_1)$ and $P=M(P)$. See also Voronoi lattice types.

In combinatorial geometry there are several problems and results relating to particular decompositions of certain sets. An example of such a problem is the Borsuk problem: Is it possible to divide any set of diameter $d$ in $\mathbf R^n$ into $n+1$ parts, each of them having diameter less than $d$? In $\mathbf R^n$ there are bounded sets for which a decomposition into a smaller number of such parts is impossible. Any decomposition determines a certain covering (cf. Covering (of a set)) and from any covering one can obtain a decomposition. Decompositions are closely connected with the illumination problem and the Hadwiger hypothesis.

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A decomposition of a space $X$ is a system $\mathfrak M$ of disjoint subsets with union $X$. The set $\mathfrak M$ becomes a topological space if one defines its open sets to be all the sets $\mathfrak N\subset\mathfrak M$ whose pre-images under the natural mapping $\mu\colon X\to\mathfrak M$ (assigning to each point $x\in X$ the unique set $M\subset\mathfrak M$ containing it) are open sets in $X$.

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A decomposition $\mathfrak M$ defines an equivalence relation in $X$ (and inversely of course). The topology on $\mathfrak M$ makes $\mathfrak M$ the quotient space of $X$ for this equivalence relation.

A decomposition is a locally finite covering of a space whose elements are canonical closed sets with disjoint open kernels.

M.I. Voitsekhovskii

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Instead of the word "decomposition" , "partition" is often used (in all three cases). However, "partition" has also other (natural) meanings, e.g. cf. Partition. See also, e.g., Measurable decomposition.