TITLE: Can anyone give me a good example of two interestingly different ordinary cohomology theories?
QUESTION [65 upvotes]: An answer to the following question would clarify my understanding of what a cohomology theory is. I know it's something that satisfies the Eilenberg-Steenrod axioms, and I know that those axioms allow you to work out quite a lot. But what sort of thing is not determined by the axioms? In particular, can someone give me a simple example of a space that has different cohomology groups with respect to two different theories? Obviously a trivial answer would be to take coefficients in different rings, so let me add the requirement that the coefficient rings should be the same. And if there's some other condition needed to make the question non-trivial, then add that in too.

REPLY [5 votes]: (Some inaccurate info appeared in the Answers above).
Different co/homology theories may behave differently on infinite CW-complexes. If two E-S theories are the same for the singleton then they are equivalent for all spaces homotopically dominated by finite polyhedra. It seems to me that for any other topological space one can construct two E-S theories (equal for singletons) which will give different groups (let's consider completely regular $T_1$-spaces only).
Surely dimension of the Hilbert cube   $C$   is infinite, while   $H^n(C)=0$   for every   $n > 0$. In the case of any Hausdorff compact space   $X$   the (topological) covering dimension, say   $n = \dim(X)$,   is equal to the cohomological dimension, meaning that there exists a closed subset   $A$   of   $X$   such that   $H^n(X\ A) \ne 0$,   but there is no such closed subset   $B$   that   $H^m(X\ B)\ne 0$   for any   $m > n$   (a few more words should be devoted to the infinite-dimensional case).
Intuitively the singular and the Cech co/homology view topological spaces in a drastically different way. What is connected for the cruder Cech theory might be not connected for the more sensitive singular theory; the former one is much more tolerant. One can say that the Cech theory applies the shape theory as its foundation rather than just homotopy theory. More precisely, all co/homology functors split into a composition of the homotopy functor and the rest; while the Cech functor does more, it splits into composition of the shape functor and the rest (while the shape functor is a composition of the homotopy functor and the rest).