Augmentation at 0-simplices and reduced singular homology

Let $X$ be a topological space and let $G$ be an abelian group. The augmentation at 0-simplices is the homomorphism ${\epsilon }_X:C_0(X;G)\to G$ defined by ${\epsilon }_X(\sigma \otimes g)=g$ for every singular $0$-simplex $\sigma :{\mathrm{\Delta }}^0\to X$, and extended linearly.

The reduced singular chain complex ${\tilde{C}}_{\bullet }(X;G)$ is obtained from the singular chain complex by replacing degree $0$ with the kernel of the augmentation: ${\tilde{C}}_n(X;G):=\{\begin{array}{ll}{C}_n(X;G),& n\ge 1,\\ \ker {\epsilon }_X,& n=0,\\ 0,& n<0.\end{array}$ Its differential is the singular boundary in degrees $\ge 1$ and the zero map ${\tilde{C}}_0(X;G)\to 0$ in degree $0$.

The reduced singular homology groups are ${\tilde{H}}_n^{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}(X;G):=H_n({\tilde{C}}_{\bullet }(X;G))\mathrm{.}$ For the empty space, this convention leaves ${\tilde{C}}_0(\mathrm{\varnothing };G)=0$, so ${\tilde{H}}_n^{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}(\mathrm{\varnothing };G)=0$ for all $n\ge 0$.