According to the continuum point-set model of solidity, all the points of any "X" ⊂ ℝ3 can be classified according to their neighborhoods with respect to "X" as interior, exterior, or boundary points. Assuming ℝ3 is endowed with the typical Euclidean metric, a neighborhood of a point "p" ∈"X" takes the form of an 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 of their interior. Any "X" ⊂ ℝ3 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 ℝ3 (by the Heine-Borel theorem it is implied that all solids are 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. The regularized set operations are denoted ∪∗, ∩∗, and −∗.