Chapter VII - WHITEHEAD CELL-COMPLEXES

Summary

Definition of a cell-complex, and the basic properties of CW-complexes. For convenience, since simplicial subdivision is a tedious operation in many cases, and for greater generality, it is advisable to extend our notion of a simplicial complex to the more general notion of a cell-complex. This extension is due to J. H. C. Whitehead, who defined a cell-complex as follows.

A cell-complex, K, is a Hausdorff space which is the union of disjoint (open) cells, eⁿ. The closure, ēⁿ, of the cell eⁿ, is the image of an n-element Eⁿ under a map f:Eⁿ, Sⁿ⁻¹→K, Kⁿ⁻¹ such that f │ Eⁿ − Sⁿ⁻¹ is a homeomorphism on to eⁿ, where Kⁿ⁻¹ is the point-set union of the cells whose dimension does not exceed (n − 1). Thus, in the terminology of Chapter VI, eⁿ is attached to Kⁿ⁻¹ by the map f│Sⁿ⁻¹ and f is a characteristic map for eⁿ. It should be noted that this definition is certainly consistent with the topology of K. For, since Eⁿ is compact and K is Hausdorff, ēⁿ is certainly a closed set. Moreover, there can be no closed set F satisfying eⁿ ⊂ F ⊂ ēⁿ, since there is no closed set f⁻¹(F) satisfying Eⁿ−Sⁿ⁻¹⊂f⁻¹(F)⊂Eⁿ, the inclusions being, of course, strict inclusions.

A subcomplex, L, of K is the union of certain cells of K, such that if eⁿ ⊂ L, then ēⁿ ⊂ L.