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In
mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It has no generally ...
, a cubical complex or cubical set is a set composed of points,
line segment 250px, The geometric definition of a closed line segment: the intersection of all points at or to the right of ''A'' with all points at or to the left of ''B'' In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' ...
s,
square In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space ...
s,
cube In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space ...
s, and their . They are used analogously to
simplicial complex In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
es and
CW complexA CW complex is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This class of spaces is broader and has some better catego ...
es in the computation of the homology of
topological space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s.

# Definitions

An elementary interval is a subset $I\subsetneq\mathbf$ of the form :
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimensio ...
$\mathbf^d$ (for some $n,d\in\mathbf\cup\$ with $n\leq d$). A set $X\subseteq\mathbf^d$ is a cubical complex (or cubical set) if it can be written as a union of elementary cubes (or possibly, is
homeomorphic and a donut (torus) illustrating that they are homeomorphic. But there need not be a continuous deformation for two spaces to be homeomorphic — only a continuous mapping with a continuous inverse function. In the mathematics, mathematical fi ...
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 $Q$, denoted $\dim Q$. The dimension of a cubical complex $X$ is the largest dimension of any cube in $X$. If $Q$ and $P$ are elementary cubes and $Q\subseteq P$, then $Q$ is a face of $P$. If $Q$ is a face of $P$ and $Q\neq P$, then $Q$ is a proper face of $P$. If $Q$ is a face of $P$ and $\dim Q=\dim P-1$, then $Q$ is a primary face of $P$.

# 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.