Definition:Concatenation (Topology)

This page or section has statements made on it that ought to be extracted and proved in a Theorem page. In particular: Extract the proofs into separate pages You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by creating any appropriate Theorem pages that may be needed. To discuss this page in more detail, feel free to use the talk page.

This article needs to be linked to other articles. In particular: $n$-cube for a start You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding these links. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{MissingLinks}} from the code.

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.