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?
There is no homology theory which satisfies the following conditions:
$H_0(X)$ is the free abelian group generated by the connected components of $X$.
The homomorphism $f_*:H_0(X)\to H_0(Y)$ induced by a continuous map $f:X\to Y$, maps a generator $[x]\in H_0(X)$ to the generator $[f(x)]\in H_0(Y)$.
The theory satisfies the Homotopy, Exactness, Excision and Dimension axioms.
Proof. Let $$X=\{(0,1)\} \cup ([0,1]\times \{0\}) \cup \bigcup\limits_{n\ge 1} (\{\frac{1}{n}\}\times [0,1]) \subseteq \mathbb{R}^2$$ with the usual subspace topology. Let $A=X\smallsetminus \{(0,1)\}$ and $U=\{(x,y)\in X \ | \ y<x\} \subseteq X$. Then $U$ and $A$ are open and $\overline{U}\subseteq int(A)=A$. By Excision $H_0(X\smallsetminus U, A\smallsetminus U)$ is isomorphic to $H_0(X,A)$. Since $A$ and $X$ are connected, $H_0(A)\to H_0(X)$ is the identity by 2. Since $A$ is contractible, by Homotopy it has the homology of a point, so by Dimension $H_{-1}(A)=0$. Then by Exactness $H_0(X,A)=0$. On the other hand $\{(0,1)\}$ is a connected component of $X\smallsetminus U$ and $(0,1)\notin A\smallsetminus U$. Therefore $H_0(A\smallsetminus U)\to H_0(X\smallsetminus U)$ is not surjective (by 1 and 2). By Exactness, $H_0(X\smallsetminus U)\to H_0(X\smallsetminus U, A\smallsetminus U)$ is not the trivial homomorphism. Then $H_0(X\smallsetminus U, A\smallsetminus U)\neq 0$, a contradiction.