Definition talk:Concatenation (Topology)

There are some issues with this definition. First, it is not compatible with Definition:Concatenation of Paths. To be able to concatenate two paths $c_1, c_2$ Definition:Concatenation (Topology) requires that:


 * $\map {c_1} {\partial \closedint 0 1^n} = {c_2} {\partial \closedint 0 1^n} = x_0$

that is $\map { c_1 } 0 = \map {c_1 } 1 = \map { c_2 } 0 = \map {c_2 } 1 = x_0$.

However, Definition:Concatenation of Paths only requires that


 * $\map {c_1 } 1 = \map { c_2 } 0 = x_0$

which appears to be the standard definition for concatenation of paths. It is used in Munkres' Topology, for instance.

I believe that the Definition:Concatenation (Topology) is designed to concatenate homotopy classes, and we should add that to the definition.

Second, I note that $c_1, c_2$ as defined here have domains equal to the $n$-cube, while $c_1 , c_2$ of Definition:Homotopy Group have domains equal to the $n$-sphere. I don't have books about the higher fundamental groups, but from what I have read on the net, the $n$-sphere appears to be the standard choice.

Any thoughts on how to resolve this? Anghel (talk) 21:47, 26 August 2022 (UTC)


 * This is an early page from before we had set in place the structure of the website, and well before we insisted that definitions be sourced from somewhere. This is of course a standard example of what happens when we rely upon the knowledge of the contributors themselves rather than requesting that they back up their "knowledge" with authority.


 * If you want to replace this definition with what's in Munkres please do so. There is nothing linking to this page, so it is not going to compromise any results which depend upon the definition being what it currently is. --prime mover (talk) 05:27, 27 August 2022 (UTC)