Algebraic Topology/Singular homology

Definition (standard n-simplex):

The standard ${\displaystyle n}$-simplex is the set

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

Definition (singular chain complex):

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

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

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