Abstract
provide a protective representation of H(X) as a direct product. It is easily verified that the singular homology and cohomology theories are additive. Also the Cech theories based on infinite coverings are additive. On the other hand James and Whitehead [4] have given examples of homology theories which are not additive. Let W~ denote the category consisting of all pairs (X, A) such that both X and A have the homotopy type of a CW-complex; and all continuous maps between such pairs. (Compare [5].) The main object of this note is to show that there is essentially only one additive homology theory and one additive cohomology theory, with given coefficient group, on the category ^ . First consider a sequence Kλ c JK2 C K3 a of C W-complexes with union K. Each K{ should be a subcomplex of K. Let H* be an additive homology theory on the category
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