Definition:Concatenation (Topology)

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Definition

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.

Also see

• Definition:Homotopy Group

Special cases

• Definition:Concatenation of Paths

Linguistic Note

The word concatenation derives from the Latin com- for with/together and the Latin word catena for chain.

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