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?