\begin{definition}[Definition:Homology Group]
Let $X$ be a topological space.
Let the standard n-simplex be denoted:
:$\ds \Delta^n := \set {\tuple {x_0, \ldots, x_n} \in {\R_{\ge 0} }^{n + 1}: \sum_{i \mathop = 1}^{n + 1} x_i = 1}$
Let $\map \CC {\Delta^n, X}$ be the set of continuous mappings from $\Delta^n$ to $X$. 
For $n \ge 0$, let $\map {C_n} X$ be the free abelian group generated by $\map \CC {\Delta^n, X}$. 
Then there is a boundary map $\partial_n: \map {C_n} X \to \map {C_{n - 1} } X$ defined as follows.
First, there are maps $s^i_n: \Delta^{n - 1} \to \Delta^n$, where $n > 0$ and $0 \le i \le n$, defined by:
:$\map {s^i_n} {x_0, \ldots, x_{n - 1} } = \tuple {x_0, \ldots, x_{i - 1}, 0, x_i, \ldots, x_{n - 1} }$
These can be considered as the inclusion of $\Delta^{n - 1}$ as a 'face' of $\Delta^n$.
For a continuous function $\phi: \Delta^n \to X$ we define:
:$\ds \map {\partial_n} \phi = \sum_{i \mathop = 0}^n \paren {-1}^i \phi \circ s^i_n$
This definition, along with the requirement that $\partial_n$ be a group homomorphism, uniquely specifies $\partial_n$.
In addition, one has $\partial_{n - 1} \partial_n = 0$, meaning that the sequence of groups and morphisms:
:$0 \gets \map {C_0} X \stackrel {\partial_1} {\longleftarrow} \map {C_1} X \stackrel {\partial_2} {\longleftarrow} \map {C_2} X \stackrel {\partial_3} {\longleftarrow} \cdots$
are a chain complex.
This is demonstrated in Singular Chains form Chain Complex.
Let $\partial_0$ denote the map $\map {C_0} X \to 0$.
Thus the $n$th singular homology group of $X$ is defined as the $n$th homology group of this chain complex.
Explicitly:
Let $\map {B_n} X \subset \map {C_n} X$ denote the image of $\partial_{n+1}$.
Let $\map {Z_n} x$ denote the kernel of $\partial_n$.
Since $\partial_n \partial_{n + 1} = 0$:
:$\map {B_n} X \subseteq \map {Z_n} X$
Then define:
:$\map {H_n} X = \dfrac {\map {Z_n} X} {\map {B_n} X}$
\end{definition}