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All manufactured components have finite size and well behaved [[Boundary (topology)|boundaries]], so initially the focus was on mathematically modeling rigid parts made of homogeneous [[isotropic]] material that could be added or removed. These postulated properties can be translated into properties of ''regions'', subsets of three-dimensional [[Euclidean space]]. The two common approaches to define "solidity" rely on ''[[point-set topology]]'' and ''[[algebraic topology]]'' respectively. Both models specify how solids can be built from simple pieces or cells.
[[File:Regularize1.png|thumb|right|450px|Regularization of a
According to the continuum point-set model of solidity, all the points of any ''X'' ⊂ ℝ<sup>3</sup> can be classified according to their ''[[Neighborhood (topology)|neighborhoods]]'' with respect to ''X'' as ''[[Interior (topology)|interior]]'', ''[[Exterior (topology)|exterior]]'', or ''[[Boundary (topology)|boundary]]'' points. Assuming ℝ<sup>3</sup> is endowed with the typical [[Euclidean metric]], a neighborhood of a point ''p'' ∈''X'' takes the form of an [[Ball (mathematics)|open ball]]. For ''X'' to be considered solid, every neighborhood of any ''p'' ∈''X'' must be consistently three dimensional; points with lower-dimensional neighborhoods indicate a lack of solidity. Dimensional homogeneity of neighborhoods is guaranteed for the class of '''closed regular sets''', defined as sets equal to the ''[[Closure (topology)|closure]]'' of their interior. Any ''X'' ⊂ ℝ<sup>3</sup> can be turned into a closed regular set or "regularized" by taking the closure of its interior, and thus the modeling space of solids is mathematically defined to be the space of closed regular subsets of ℝ<sup>3</sup> (by the [[Heine–Borel theorem|Heine-Borel theorem]] it is implied that all solids are [[Compact space|compact]] sets). In addition, solids are required to be closed under the Boolean operations of set union, intersection, and difference (to guarantee solidity after material addition and removal). Applying the standard Boolean operations to closed regular sets may not produce a closed regular set, but this problem can be solved by regularizing the result of applying the standard Boolean operations.<ref name = "Regularized operations">{{citation|doi=10.1016/0010-4485(80)90025-1|title=Closure of Boolean operations on geometric entities|journal=Computer-Aided Design|volume=12|issue=5|pages=219–220|year=1980|last1=Tilove|first1=R.B.|last2=Requicha|first2=A.A.G.}}</ref> The regularized set operations are denoted ∪<sup>∗</sup>, ∩<sup>∗</sup>, and −<sup>∗</sup>.
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