In singular homology one of the first calculations you can make is $H_0(X)=H_0(pt)$ for path-connected $X$. This seems to be a property which does not follow from the axioms for a generalized homology theory. This raises the following question:

Assume $H_* : Top^2 \to Ab^\mathbb{N}$ is a homology theory. Thus we impose homotopy invariance, excision, the long exact sequence, the dimension axiom and if you wish also the disjoint union axiom. Is then $H_0(X)=H_0(pt)$ for every path-connected space $X$?

I believe that there is a counterexample. Of course this can't be homotopy equivalent to a CW-complex. And probably this is the reason why this question is not reasonable at all. It's just my curiosity.

homologyhas at least three definitions: one is the inverse limit of homology groups over open covers (Aleksandrov–Cech homology), one takes values in pro-abelian groups, and one applies a derived version of the inverse limit functor to the chain complexes in the system of open covers (the version I was using). I'm not sure of a reference and I would have to check closely to see if it satisfies the other axioms. One has to define the relative Cech homology using a mapping cylinder so that the long exact sequence becomes a tautology. – Tyler Lawson Oct 7 '10 at 14:08