CW complex
A Hausdorff topological space $X$ is said to be a CW complex if it the following conditions:

1.
There exists a filtration^{} by subspaces^{}
$${X}^{(1)}\subseteq {X}^{(0)}\subseteq {X}^{(1)}\subseteq {X}^{(2)}\subseteq \mathrm{\cdots}$$ with $X=\bigcup _{n\ge 1}{X}^{(n)}.$

2.
${X}^{(1)}$ is empty, and, for $n\ge 0,{X}^{(n)}$ is obtained from ${X}^{(n1)}$ by attachment of a collection^{} $\{{e}_{\iota}^{n}:\iota \in {I}_{n}\}$ of $n$cells.

3.
(“closurefinite”) Every closed cell is contained in a finite union of open cells.

4.
(“weak topology”) $X$ has the weak topology with respect to the collection of all cells. That is, $A\subset X$ is closed in $X$ if and only if the intersection^{} of $A$ with every closed cell $e$ is closed in $e$ with respect to the subspace topology.
The letters ‘C’ and ‘W’ stand for “closurefinite” and “weak topology,” respectively. In particular, this means that one shouldn’t look too closely at the initials of J.H.C. Whitehead, who invented CW complexes.
The subspace ${X}^{(n)}$ is called the $n$skeleton of $X.$ Note that there normally are many possible choices of a filtration by skeleta for a given CW complex. A particular choice of skeleta and attaching maps for the cells is called a CW structure^{} on the space.
Intuitively, $X$ is a CW complex if it can be constructed, starting from a discrete space, by first attaching onecells, then twocells, and so on. Note that the definition above does not allow one to attach $k$cells before $h$cells if $k>h.$ While some authors allow this in the definition, it seems to be common usage to restrict CW complexes to the definition given here, and to call a space constructed by cell attachment with unrestricted order of dimensions^{} a cell complex. This is not essential for homotopy^{} purposes, since any cell complex is homotopy equivalent to a CW complex.
CW complexes are a generalization^{} of simplicial complexes^{}, and have some of the same advantages. In particular, they allow inductive reasoning on the of skeleta. However, CW complexes are far more flexible than simplicial complexes. For a space $X$ drawn from “everyday” topological spaces^{}, it is a good bet that it is homotopy equivalent, or even homeomorphic^{}, to a CW complex. This includes, for instance, smooth finitedimensional manifolds, algebraic varieties, certain smooth infinitedimensional manifolds (such as Hilbert manifolds), and loop spaces^{} of CW complexes. This makes the category^{} of spaces homotopy equivalent to a CW complex a very popular category for doing homotopy theory.
Remark 1.
There is potential for confusion in the way words like “open” and “interior” are used for cell complexes. If ${e}^{k}$ is a closed $k$cell in CW complex $X$ it does not follow that the corresponding open cell ${\stackrel{\mathrm{\circ}}{e}}^{k}$ is an open set of $X\mathrm{.}$ It is, however, an open set of the $k$skeleton. Also, while ${\stackrel{\mathrm{\circ}}{e}}^{k}$ is often referred to as the “interior” of ${e}^{k}\mathrm{,}$ it is not necessarily the case that it is the interior of ${e}^{k}$ in the sense of pointset topology. In particular, any closed $\mathrm{0}$cell is its own corresponding open $\mathrm{0}$cell, even though it has empty interior in most cases.
Title  CW complex 
Canonical name  CWComplex 
Date of creation  20130322 13:26:02 
Last modified on  20130322 13:26:02 
Owner  antonio (1116) 
Last modified by  antonio (1116) 
Numerical id  10 
Author  antonio (1116) 
Entry type  Definition 
Classification  msc 57XX 
Classification  msc 55XX 
Synonym  CWcomplex 
Related topic  SimplicialComplex 
Related topic  CellAttachment 
Related topic  ApproximationTheoremForAnArbitrarySpace 
Related topic  SpinNetworksAndSpinFoams 
Related topic  CWComplexDefinitionRelatedToSpinNetworksAndSpinFoams 
Related topic  GeneralizedHurewiczFundamentalTheorem 
Related topic  VariableTopology3 
Related topic  QuantumAlgebraicTopologyOfCWComplexRepres 
Defines  skeleton 
Defines  skeleta 
Defines  closurefinite 
Defines  cell complex 
Defines  CW structure 
Defines  CWstructure 