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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
I would comment this, but don't have points to do so. Does Alexander-Spanier cohomology work? A reference would be Massey's book on homology and cohomology. –  Kyle Dec 31 '14 at 17:05

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