The augmentation map $\epsilon:S_0(X)\longrightarrow \mathbb{Z}\;$:

Since the standard Euclidean simplex $\Delta_0$ is a singleton, each singular zero simplex $\Delta_0\longrightarrow X$ can be identified with a point of $X$ namely the image of the singular zero simplex. Thus we may think of a singular zero chain as an element of the free abelian group generated by the points of $X$, that is a formal expression

$$c_1p_1 + c_2p_2 + \dots + c_kp_k, \eqno(31.1)$$

where $p_1, p_2,\dots, p_k$ are points of $X$ and the coefficients $c_1, c_2,\dots, c_k$ are integers.