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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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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
What exactly do you mean by « a broad sense»? Does the free abelian group on the set of connected components count? – Mariano Suárez-Alvarez Apr 30 '15 at 4:38
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 '15 at 17:37

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