Cubical complex

An elementary interval is a subset ${\displaystyle I\subsetneq \mathbf {R} }$ of the form

${\displaystyle I=[l,l+1]\quad {\text{or}}\quad I=[l,l]}$

for some ${\displaystyle l\in \mathbf {Z} }$. An elementary cube ${\displaystyle Q}$ is the finite product of elementary intervals, i.e.

${\displaystyle Q=I_{1}\times I_{2}\times \cdots \times I_{d}\subsetneq \mathbf {R} ^{d}}$

where ${\displaystyle I_{1},I_{2},\ldots ,I_{d}}$ are elementary intervals. Equivalently, an elementary cube is any translate of a unit cube ${\displaystyle [0,1]^{n}}$ embedded in Euclidean space ${\displaystyle \mathbf {R} ^{d}}$ (for some ${\displaystyle n,d\in \mathbf {N} \cup \{0\}}$ with ${\displaystyle n\leq d}$).[2] A set ${\displaystyle X\subseteq \mathbf {R} ^{d}}$ 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]

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.

• Simplicial complex
• Simplicial homology
• Abstract cell complex