A CW complex (also called cellular complex or cell complex) is a kind of a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
that is particularly important in
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
. 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
categorical properties than
simplicial complexes, but still retains a combinatorial nature that allows for computation (often with a much smaller complex). The ''C'' stands for "closure-finite", and the ''W'' for "weak" topology.
Definition
CW complex
A CW complex is constructed by taking the union of a sequence of topological spaces
such that each
is obtained from
by gluing copies of k-cells
, each homeomorphic to
, to
by continuous gluing maps
. The maps are also called
attaching maps.
Each
is called the k-skeleton of the complex.
The topology of
is weak topology: a subset
is open iff
is open for each cell
.
In the language of category theory, the topology on
is the
direct limit of the diagram
The name "CW" stands for "closure-finite weak topology", which is explained by the following theorem:
This partition of ''X'' is also called a cellulation.
The construction, in words
The CW complex construction is a straightforward generalization of the following process:
* A 0-''dimensional CW complex'' is just a set of zero or more discrete points (with the
discrete topology
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are ''isolated'' from each other in a certain sense. The discrete topology is the finest top ...
).
* A 1-''dimensional CW complex'' is constructed by taking the
disjoint union
In mathematics, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injection of each A_i into A, such that the images of these injections form a partition of A ( ...
of a 0-dimensional CW complex with one or more copies of the
unit interval
In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysis ...
. For each copy, there is a map that "
glues" its boundary (its two endpoints) to elements of the 0-dimensional complex (the points). The topology of the CW complex is the topology of the
quotient space defined by these gluing maps.
* In general, an ''n-dimensional CW complex'' is constructed by taking the disjoint union of a ''k''-dimensional CW complex (for some