Prove $H_0 (X,A)=\bigoplus H_0 (X_i,X_i\cap A)$ for $X_i$ the path components Let $X$ be a topological spcae, $X_i$ its path components, and $A\subset X$ a subspace.
I'm interested in proving $H_0(X,A)=\bigoplus H_0(X_i,X_i\cap A)$.
my work by now (I have completely proved the following equality):
$H_0(X,A)=\frac{\frac{C_0(X)}{C_0(A)}}{im\partial_1}=\frac{\frac{\bigoplus C_0(X_i)}{\bigoplus C_0(X_i\cap A)}}{im\partial_1}=\frac{\bigoplus \frac{C_0(X_i)}{C_0(X_i\cap A)}}{im\partial_1}$.
now to prove that $H_0(X,A)=\bigoplus H_0(X_i,X_i\cap A)$ I wanted to define a homomorphism $f:\bigoplus \frac{C_0(X_i)}{C_0(X_i\cap A)}\to \bigoplus H_0(X_i,X_i\cap A)$ which is a surejection, and has kernel $im\partial_1$, and then use the 1st homomorphism theorem.
There is a natural way of doing that, but I couldn't show that it satisfices the desired properties.
Could you help me finishing the argument? (or propose a different way)
 A: We have inclusions $(X_i,A\cap X_i)\hookrightarrow(X,A)$, inducing a map $H_n(X_i,A\cap X_i)\to H_n(X,A)$, and these maps together give a map $\iota:\oplus H_n(X_i,A\cap X_i)\to H_n(X,A)$. On the other hand, given a relative $n$-cycle $\gamma+C_n(A)$, the chain $\gamma$ can be expressed as $\gamma=\sum_i \gamma_i$, where $\gamma_i$ is the sum of those simplices of $\gamma$ lying in $X_i$. Then $\gamma_i$ must have its boundary in $C_{n-1}(A\cap X_i)$, so it is a relative cycle and thus represents an element $[\gamma_i+C_n(A\cap X_i)]\in H_n(X_i,A\cap X_i)$. This is basically the map $Z_n(X,A)\to \oplus Z_n(X_i,A\cap X_i)$, which you already thought about in dimension $0$ where $Z_0=C_0$.
In order to show that this map factors through $H_n(X,A)$, we need to show that a relative boundary, that is a coset $\beta+C_n(A)$, where $\beta=\partial\alpha$ for some $\alpha\in C_{n+1}(X)$, is mapped to $0$ in the direct sum. But $\partial\alpha=\beta$ implies that $\partial\alpha_i=\beta_i$, so $\sum_i\beta_i$ lies in $\oplus_i B_n(X_i)$.  
We therefore have a map $\rho:H_n(X,A)\to\oplus H_n(X_i,A\cap X_i)$. Can you show that $\iota$ and $\rho$ are inverse to each other?
