\begin{definition}[Definition:Concatenation (Topology)]
Let $T = \struct {S, \tau}$ be a topological space.
Let $c_1, c_2: \closedint 0 1^n \to S$ be maps.
Let $c_1$ and $c_2$ both satisfy the concatenation criterion:
:$\map c {\partial \closedint 0 1^n} = x_0$.
Then the '''concatenation''' $c_1 * c_2$ is defined as:
:$\map {\paren {c_1 * c_2} } {t_1, t_2, \ldots, t_n} = \begin {cases}
\map {c_1} {2 t_1, t_2, \ldots, t_n} & : t_1 \in \closedint 0 {1/2} \\ 
\map {c_2} {2 t_1 - 1, t_2, \ldots, t_n} & : t_1 \in \closedint {1/2} 1
\end{cases} $ 
where $\tuple {t_1, \ldots, t_n}$ are coordinates in the $n$-cube.
By Continuous Mapping on Finite Union of Closed Sets, $c_1 * c_2$ is continuous.
This resulting map is continuous, since:
:$2 \paren {\dfrac 1 2} = 1$ and $2 \paren {\dfrac 1 2} - 1 = 0$
:anywhere any coordinate of $\hat t$ is either $1$ or $0$, $\map {\paren {c_1 * c_2} } {\hat t} = x_0$.
The resulting map also clearly satisfies the concatenation criteria itself.
\end{definition}