Algebraic Topology/Singular homology

Definition (standard n-simplex):

The standard $n$-simplex is the set

$\Delta _{n}:=\left\{(t_{0},\ldots ,t_{n}){\big |}\forall 0\leq j\leq n:t_{j}\geq 0\wedge \sum _{j=0}^{n}t_{j}=1\right\}$.

Definition (singular chain complex):

Let $X$ be a topological space. The singular chain complex associated to $X$ is the chain complex $C_{n}(X)$ of abelian groups whose $n$-th group is given by the free abelian group over all continuous functions $\sigma :\Delta _{n}\to X$ and whose differential $\partial :C_{n}(X)\to C_{n-1}(X)$ is given by the linear extension of the formula

$\partial \sigma =\sum _{j=0}^{n}(-1)^{j}\sigma _{j}$ for $\sigma :\Delta _{n}\to X$ continuous,

where $\sigma _{j}:\Delta _{n-1}\to X,\sigma _{j}(t_{0},\ldots ,t_{n-1}):=\sigma (t_{0},\ldots ,t_{j-1},0,t_{j},\ldots ,t_{n-1})$.