If $\{X_{\lambda} : \lambda \in \Gamma\}$ is the set of path components of $X$, then for every $n \ge 0$, $$H_n(X) \approx \sum_{\lambda}H_n(X_{\lambda})$$ where $H_n(X)$ is the singular homology group of $X$ and the summation sign means direct sum.
I figure that I have to show that $\theta_n: \text{cls } \gamma \mapsto \text{cls } \gamma_{\lambda}$ is an isomorphism where $\gamma = \sum_{\lambda} \gamma_{\lambda}$, but I'm having trouble showing that it's well defined as the notation becomes very cumbersome as the amount of indexing increases.
$$\sum_{\lambda}(\sum_im_{(i,\lambda)}\sigma_{(i,\lambda)}) + B_n(X) = \sum_{\lambda}(\sum_iv_{(i,\lambda)}\sigma_{(i,\lambda)}) + B_n(X)$$ $$\Rightarrow$$ $$\sum_im_{(i,\lambda)}\sigma_{(i,\lambda)} + B_n(X_{\lambda}) = \sum_iv_{(i,\lambda)}\sigma_{(i,\lambda)} + B_n(X_{\lambda})$$
Anyone have any ideas how to show this is well defined? I'm self learning this, so any help would be appreciated.