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Fernando Muro
Fernando Muro
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Hatcher's Theorem 4E.1 in his 'Algebraic Topology' book is equivalent to Brown's Theorem II version (4.6) in his 'Cohomology theories' paper. Countability of coefficients is only required for Theorem II version (4.5). The first theorem is for cohomology theories defined on all CW-complexes, and the second one for those defined on finite CW-complexes. It looks like if the countability hypothesis were necessary if we wantwanted to represent cohomology theories defined only on finite CW-complexes, but this hypothesis was removed by Adams in 'A variant of E. H. Brown's representability theorem'.

Hatcher's Theorem 4E.1 in his 'Algebraic Topology' book is equivalent to Brown's Theorem II version (4.6) in his 'Cohomology theories' paper. Countability of coefficients is only required for Theorem II version (4.5). The first theorem is for cohomology theories defined on all CW-complexes, and the second one for those defined on finite CW-complexes. It looks like if the countability hypothesis were necessary if we want to represent cohomology theories defined only on finite CW-complexes, but this hypothesis was removed by Adams in 'A variant of E. H. Brown's representability theorem'.

Hatcher's Theorem 4E.1 in his 'Algebraic Topology' book is equivalent to Brown's Theorem II version (4.6) in his 'Cohomology theories' paper. Countability of coefficients is only required for Theorem II version (4.5). The first theorem is for cohomology theories defined on all CW-complexes, and the second one for those defined on finite CW-complexes. It looks like if the countability hypothesis were necessary if we wanted to represent cohomology theories defined only on finite CW-complexes, but this hypothesis was removed by Adams in 'A variant of E. H. Brown's representability theorem'.

Source Link
Fernando Muro
Fernando Muro
  • 15.2k
  • 2
  • 49
  • 78

Hatcher's Theorem 4E.1 in his 'Algebraic Topology' book is equivalent to Brown's Theorem II version (4.6) in his 'Cohomology theories' paper. Countability of coefficients is only required for Theorem II version (4.5). The first theorem is for cohomology theories defined on all CW-complexes, and the second one for those defined on finite CW-complexes. It looks like if the countability hypothesis were necessary if we want to represent cohomology theories defined only on finite CW-complexes, but this hypothesis was removed by Adams in 'A variant of E. H. Brown's representability theorem'.