Math 2.0

Manifolds and cell complexes Definition

CW complex

Builds on Hausdorff spaceClosed ballHomeomorphismContinuous map of topological spaces

Definition

Let XX be a Hausdorff space. For n≥0n \ge 0 let DnD^n be the closed unit ball in ℝn\R^n, with interior D˚n\mathring{D}^n and boundary Sn−1S^{n-1}. A CW structure on XX is a partition of XX into subsets eαe_\alpha, called cells, with extra data. Each cell has a dimension nα≥0n_\alpha \ge 0 and a continuous map Φα ⁣:Dnα→X\Phi_\alpha \colon D^{n_\alpha} \to X, its characteristic map. These data satisfy:

  1. Φα\Phi_\alpha restricts to a homeomorphism from D˚nα\mathring{D}^{n_\alpha} onto eαe_\alpha.
  2. Φα(Snα−1)\Phi_\alpha(S^{n_\alpha - 1}) is contained in the union of finitely many cells eβe_\beta with nβ<nαn_\beta < n_\alpha (closure finiteness).
  3. A subset A⊆XA \subseteq X is closed if and only if Φα−1(A)\Phi_\alpha^{-1}(A) is closed in DnαD^{n_\alpha} for every α\alpha (weak topology).

A CW complex is a Hausdorff space with a CW structure. The cell eαe_\alpha is an nαn_\alpha-cell, and the union XnX^n of all cells of dimension at most nn is the nn-skeleton.

Remarks

Equivalently, a CW complex is built inductively. The skeleton X0X^0 is a discrete space, and XnX^n is obtained from Xn−1X^{n-1} by attaching nn-cells along maps Sn−1→Xn−1S^{n-1} \to X^{n-1}. The union X=⋃nXnX = \bigcup_n X^n has the topology in which a set is closed when its intersection with each XnX^n is closed.

Example: the sphere SnS^n with n≥1n \ge 1 has a CW structure with one 00-cell {p}\set{p} and one nn-cell Sn∖{p}S^n \setminus \set{p}. The characteristic map Dn→SnD^n \to S^n collapses Sn−1S^{n-1} to pp and is a homeomorphism on the interior, so clause 1 holds. Clause 2 holds since Φ(Sn−1)={p}\Phi(S^{n-1}) = \set{p}, and clause 3 holds since SnS^n is a quotient of Dn⊔{p}D^n \sqcup \set{p}.

Non-example: the closed disk D2D^2 with one 00-cell at each point of the boundary circle and one 22-cell, the interior, fails clause 2. The boundary of the 22-cell meets infinitely many 00-cells.

A CW complex is finite if it has finitely many cells; a finite CW complex is compact.

Reading path

To understand this page from the ground up, read these 37 pages in order. The foundations come first.

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Element, definition, no evidence levelElementdefinitionProposition, concept, no evidence levelPropositionconceptReal numbers, object, no evidence levelReal numbersobjectSet, definition, no evidence levelSetdefinitionConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionEmpty set, object, no evidence levelEmpty setobjectFamily of sets, definition, no evidence levelFamily of setsdefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionIntersection, definition, no evidence levelIntersectiondefinitionUnion, definition, no evidence levelUniondefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionUnion of a family of sets, definition, no evidence levelUnion of a family of …definitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionTopological space, definition, no evidence levelTopological spacedefinitionFunction, definition, no evidence levelFunctiondefinitionCodomain, definition, no evidence levelCodomaindefinitionInjective function, definition, no evidence levelInjective functiondefinitionPreimage of a set, definition, no evidence levelPreimage of a setdefinitionMetric space, definition, no evidence levelMetric spacedefinitionSurjective function, definition, no evidence levelSurjective functiondefinitionClosed ball, definition, no evidence levelClosed balldefinitionOpen ball, definition, no evidence levelOpen balldefinitionBijective function, definition, no evidence levelBijective functiondefinitionMetric topology, definition, no evidence levelMetric topologydefinitionInverse function, definition, no evidence levelInverse functiondefinitionOpen set, definition, no evidence levelOpen setdefinitionHausdorff space, definition, no evidence levelHausdorff spacedefinitionContinuous map of topological spaces, definition, no evidence levelContinuous map of top…definitionHomeomorphism, definition, no evidence levelHomeomorphismdefinitionCW complex, definition, no evidence levelCW complexdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Element definition
  2. Proposition concept
  3. Real numbers object
  4. Set definition
  5. Conjunction definition
  6. Disjunction definition
  7. Implication definition
  8. Predicate definition
  9. Empty set object
  10. Family of sets definition
  11. Ordered pair definition
  12. Intersection definition
  13. Union definition
  14. Existential quantifier definition
  15. Universal quantifier definition
  16. Cartesian product definition
  17. Subset definition
  18. Union of a family of sets definition
  19. Uniqueness quantifier definition
  20. Binary relation definition
  21. Topological space definition
  22. Function definition
  23. Codomain definition
  24. Injective function definition
  25. Preimage of a set definition
  26. Metric space definition
  27. Surjective function definition
  28. Closed ball definition
  29. Open ball definition
  30. Bijective function definition
  31. Metric topology definition
  32. Inverse function definition
  33. Open set definition
  34. Hausdorff space definition
  35. Continuous map of topological spaces definition
  36. Homeomorphism definition
  37. CW complex definition

Further reading

  1. Web https://en.wikipedia.org/wiki/CW_complex
About this page
Tag5SP8E66T9E
URIm2://5SP8E66T9E
Kinddefinition
WritingW0 stub
MSC57Q05, 55U10
DefinesCW complex
UsesHausdorff space, Closed ball, Homeomorphism, Continuous map of topological spaces
Changed2 November 2026

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