# CW complex

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

1. 1.
 ${X}^{(-1)}\subseteq{X}^{(0)}\subseteq{X}^{(1)}\subseteq{X}^{(2)}\subseteq\cdots$

with $X=\bigcup\limits_{n\geq-1}{X}^{(n)}.$

2. 2.

${X}^{(-1)}$ is empty, and, for $n\geq 0,{X}^{(n)}$ is obtained from ${X}^{(n-1)}$ by attachment of a collection  ${\left\{e_{\iota}^{n}:\>\iota\in I_{n}\right\}}$ of $n$-cells.

3. 3.

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

4. 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 “closure-finite” 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 on the space.

Intuitively, $X$ is a CW complex if it can be constructed, starting from a discrete space, by first attaching one-cells, then two-cells, 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 finite-dimensional manifolds, algebraic varieties, certain smooth infinite-dimensional 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 $\smash{\overset{\circ}{e}}^{k}$ is an open set of $X.$ It is, however, an open set of the $k$-skeleton. Also, while $\smash{\overset{\circ}{e}}^{k}$ is often referred to as the “interior” of $e^{k},$ it is not necessarily the case that it is the interior of $e^{k}$ in the sense of pointset topology. In particular, any closed $0$-cell is its own corresponding open $0$-cell, even though it has empty interior in most cases.

 Title CW complex Canonical name CWComplex Date of creation 2013-03-22 13:26:02 Last modified on 2013-03-22 13:26:02 Owner antonio (1116) Last modified by antonio (1116) Numerical id 10 Author antonio (1116) Entry type Definition Classification msc 57-XX Classification msc 55-XX Synonym CW-complex 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 closure-finite Defines cell complex Defines CW structure Defines CW-structure