We know that for sufficiently nice spaces, (e.g. spaces with the homotopy type of a CW complex) the Eilenberg-Steenrod axioms determine the ordinary cohomology of the space. One can construct nasty spaces, like the Topologist's Sine Curve, for which singular and Cech cohomology disagree.

My question: are there other families of spaces that have their ordinary cohomology determined by the Eilenberg-Steenrod axioms, or are we limited to those with the homotopy type of a CW complex?

  • $\begingroup$ I'll confess that I don't know much about this field, however, you might be interested in the topic of generalized cohomology theories. It addresses your question indirectly by giving many examples of different cohomology theories and the families of spaces to which each is adapted. But it does not address your question directly because many of these families of spaces are more restrictive than CW complexes, rather than less restrictive which seems to be what you are asking about. $\endgroup$ – Lee Mosher Oct 30 '18 at 13:51

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