Cubical complex

In mathematics, a cubical complex or cubical set is a set composed of points, line segments, squares, cubes, and their "n"-dimensional counterparts. They are used analogously to simplicial complexes and CW complexes in the computation of the homology of topological spaces.
Definitions.
An elementary interval is a subset formula_1 of the form
for some formula_3. An elementary cube formula_4 is the finite product of elementary intervals, i.e.
where formula_6 are elementary intervals. Equivalently, an elementary cube is any translate of a unit cube formula_7 embedded in Euclidean space formula_8 (for some formula_9 with formula_10). A set formula_11 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).
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 formula_4, denoted formula_13. The dimension of a cubical complex formula_14 is the largest dimension of any cube in formula_14.
If formula_4 and formula_17 are elementary cubes and formula_18, then formula_4 is a face of formula_17. If formula_4 is a face of formula_17 and formula_23, then formula_4 is a proper face of formula_17. If formula_4 is a face of formula_17 and formula_28, then formula_4 is a primary face of formula_17.
Algebraic topology.
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.