CW complex

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

  1. 1.

    There exists a filtrationPlanetmathPlanetmath by subspacesMathworldPlanetmath


    with X=n-1X(n).

  2. 2.

    X(-1) is empty, and, for n0,X(n) is obtained from X(n-1) by attachment of a collectionMathworldPlanetmath {eιn:ιIn} 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, AX is closed in X if and only if the intersectionMathworldPlanetmathPlanetmath 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 CW structureMathworldPlanetmath 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 dimensionsMathworldPlanetmath a cell complex. This is not essential for homotopyMathworldPlanetmath purposes, since any cell complex is homotopy equivalent to a CW complex.

CW complexes are a generalizationPlanetmathPlanetmath of simplicial complexesMathworldPlanetmath, 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 spacesMathworldPlanetmath, it is a good bet that it is homotopy equivalent, or even homeomorphicMathworldPlanetmath, 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 spacesMathworldPlanetmath of CW complexes. This makes the categoryMathworldPlanetmath 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 ek is a closed k-cell in CW complex X it does not follow that the corresponding open cell ek is an open set of X. It is, however, an open set of the k-skeleton. Also, while ek is often referred to as the “interior” of ek, it is not necessarily the case that it is the interior of ek 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