# Is there a homology theory that counts connected components of a space?

It is well-known that the generators of the zeroth singular homology group $H_0(X)$ of a space $X$ correspond to the path components of $X$.

I have recently learned that for Čech homology the corresponding statement would be that $\check{H}_0(X)$ is generated by the quasicomponents of $X$. This leads me to my question:

Are there any homology theories (in a broad sense; i.e. not necessarily satisfying all of Eilenberg-Steenrod axioms) being used such that the zeroth homology of a space is generated by its connected components?

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(my answer is incorrect, I was forgetting my basic point-set nonsense: components don't have to be clopen... Sorry!) –  Dylan Wilson Mar 2 '13 at 6:46
Does 0'th sheaf cohomology (with constant coefficients) count components or quasicomponents? –  Grigory M Jul 3 '13 at 18:47
$H_{0}(PX)$, where $PX =$ the path space of X with compact open topology. I would have rather liked to put it as a comment but I do not have enough points to do so. –  DBS Jul 7 '13 at 22:32
@GrigoryM quasicomponents: if $p,q\in X$ are in the same quasicomponent, they can't be divided by a global section of a locally constant sheaf. –  Giulio Bresciani Dec 22 '14 at 21:21
That is not what I mean. Define $H_0(X)$ to be the free abelian group on the components of $X$ and let $H_0(X,Y)$ and $H_1(X,Y)$ be the cokernel and kernel of the map $X_0(Y)\to H_0(X)$. Let $H_p=0$ for all $p>1$. This has long exact sequences for pairs, is additive and satisfies the dimension axiom. I have no idea about excision and homotopy, but I guess they are not satisfied. I don't know how reasonable is to ask for these for a theory having components in degree zero, though — maybe one can prove these two cannot be satisfied? –  Mariano Suárez-Alvarez Apr 30 at 17:37