Definition:Concatenation (Topology)

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Definition

Let $T = \left({S, \tau}\right)$ be a topological space.

Let $c_1, c_2: \left[{0 \,.\,.\, 1}\right]^n \to S$ be maps.

Let $c_1$ and $c_2$ both satisfy the concatenation criterion $c \left({\partial \left[{0 \,.\,.\, 1}\right]^n}\right) = x_0$.


Then the concatenation $c_1 * c_2$ is defined as:

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

where $\left({t_1, \ldots, t_n}\right)$ 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:

  • $2 \left({\dfrac 1 2}\right) = 1$ and $2 \left({\dfrac 1 2}\right) - 1 = 0$;
  • anywhere any coordinate of $\hat t$ is either $1$ or $0$, $\left({c_1*c_2}\right) \left({\hat t}\right) = x_0$.

The resulting map also clearly satisfies the concatenation criteria itself.


Also see


Special cases


Linguistic Note

The word concatenation derives from the Latin word catena for chain.

However, the end result of such an operation is not to be confused with a (set theoretical) chain.