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.
An elementary interval is a subset of the form
for some . An elementary cube is the finite product of elementary intervals, i.e.
where are elementary intervals. Equivalently, an elementary cube is any translate of a unit cube embedded in Euclidean space (for some with ). A set 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).
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 , denoted . The dimension of a cubical complex is the largest dimension of any cube in .
If and are elementary cubes and , then is a face of . If is a face of and , then is a proper face of . If is a face of and , then is a primary face of .
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.
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