Reduced versus non-reduced homology

Ordinary singular homology uses the chain complex \[ \cdots\to C_1(X)\xrightarrow{\partial_1}C_0(X)\to0. \] Reduced homology replaces the bottom by the augmentation map \[ \varepsilon:C_0(X)\to\mathbb Z, \qquad \varepsilon\!\left(\sum n_i x_i\right)=\sum n_i, \] forming \[ \cdots\to C_1(X)\to C_0(X)\xrightarrow{\varepsilon}\mathbb Z\to0. \] The resulting groups are \(\widetilde H_n(X)\).

For every nonempty space, \[ \widetilde H_n(X)\cong H_n(X)\qquad(n>0), \] while degree \(0\) has the extra free summand \[ H_0(X)\cong\widetilde H_0(X)\oplus\mathbb Z. \] Thus if \(X\) has \(c\) path components, \[ H_0(X)\cong\mathbb Z^c, \qquad \widetilde H_0(X)\cong\mathbb Z^{c-1}. \] In particular, reduced homology of a point is zero in every degree.