# Why is homology isomorphic to reduced homology plus Z?

by redbowlover
Tags: homology, isomorphic, reduced
 P: 16 Working through Hatcher... For any space X, we have an augmented chain complex $...\rightarrow C_1(X) \rightarrow C_0(X)\rightarrow \mathbb{Z}\stackrel{\epsilon}{\rightarrow}0$ Hathcer says that since $\epsilon$ induces a map $H_0(X)\rightarrow \mathbb{Z}$ with kernel $\tilde{H}_0(X)$, we get an isomorphism $H_0(X)\simeq \tilde{H}_0(X)\oplus \mathbb{Z}$ Where is this isomorphism coming from? I understand where the induced map on $H_0(X)$ comes from... Thanks