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 points ...
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 invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
. It was introduced by
J. H. C. Whitehead[ (open access)] to meet the needs of
homotopy theory
In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topolog ...
. This class of spaces is broader and has some better
categorical properties than
simplicial complex
In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set ...
es, 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
In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any categor ...
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 to ...
).
* 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 (th ...
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