TITLE: The 0th homology of a path-connected space
QUESTION [8 upvotes]: 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.

REPLY [5 votes]: I'm not sure exactly what Cech homology is, but I'll assume that something which may or may not be called Cech homology has the following properties:
It's a generalized homology theory.
It vanishes in negative dimensions.
It satisfies the dimension axiom (so it's a homology theory).
There's a natural map from singular to Cech that is part of a natural triangle, an exact sequence
$\dots \to Cech_{n+1}\to ?_n\to Sing_n\to Cech_n\to \dots$.
The map $Sing_0\to Cech_0$ is always surjective, so that $?_n$ vanishes for $n<0$. 
The map $Sing_0(TSC)\to Cech_0(TSC)$ is not injective if TSC is the topologist's sine curve.
(End of list of assumed properties.)
Then $?$ is a generalized homology theory vanishing in negative dimensions and vanishing on a point (therefore on CW complexes) but not vanishing on the path-connected space that you get by attaching a suitable $1$-cell to TSC.
The direct sum of ? and singular will then do the job, because $?_0(TSC\cup cell)=?_0(TSC)\ne 0$