Definition:Concatenation (Topology)

For two maps $$c_1,c_2:[0,1]^n \to X$$ which satisfy the concatenation criteria $$c(\partial [0,1]^n)=x_0$$ holds, the concatenation $$c_1 * c_2$$ is defined as

$$(c_1 * c_2)(t_1, t_2, ... t_n) = \begin{cases} c_1(2t_1, t_2, ... t_n), & \mbox{if } t_1 \in [0,1/2]  \\ c_2(2t_1-1, t_2, ... t_n),  & \mbox{if } t_1 \in [1/2,1] \end{cases} $$

where $$(t_1, ... ,t_n) \ $$ are coordinates in the n-cube.

This resulting map is continuous, since $$2(\tfrac{1}{2}) = 1$$ and $$2(\tfrac{1}{2})-1=0$$, and anywhere any co-ordinate of $$\hat{t}$$ is either 1 or 0, $$(c_1*c_2)(\hat{t})=x_0$$. The resulting map also clearly satisfies the concatenation criteria itself.