where <math>I_1,I_2,\ldots,I_d</math> are elementary intervals. Equivalently, an elementary cube is any translate of a unit cube <math>[0,1]^n</math> embedded in Euclidean space <math>\mathbf{R}^d</math> (for some <math>n,d\in\mathbf{N}\cup\{0\}</math> with <math>n\leq d</math>).[2] A set <math>X\subseteq\mathbf{R}^d</math> is a cubical complex (or cubical set) if it can be written as a union of elementary cubes (or possibly, is homeomorphic to such a set).[3]
Related terminology
Elementary intervals of length 0 (containing a single point) are called degenerate, while those of length 1 are nondegenerate. The dimension of a cube is the number of nondegenerate intervals in <math>Q</math>, denoted <math>\dim Q</math>. The dimension of a cubical complex <math>X</math> is the largest dimension of any cube in <math>X</math>.
If <math>Q</math> and <math>P</math> are elementary cubes and <math>Q\subseteq P</math>, then <math>Q</math> is a face of <math>P</math>. If <math>Q</math> is a face of <math>P</math> and <math>Q\neq P</math>, then <math>Q</math> is a proper face of <math>P</math>. If <math>Q</math> is a face of <math>P</math> and <math>\dim Q=\dim P-1</math>, then <math>Q</math> is a facet or primary face of <math>P</math>.
In algebraic topology, cubical complexes are often useful for concrete calculations. In particular, there is a definition of homology for cubical complexes that coincides with the singular homology, but is computable.
