The definition of the product of 2 paths, according to Ralph H. Fox and Richard H. Crowell, is as follows:
Consider any two paths $a$ and $b$ in $X$ which are such that the terminal point of $a$ coincides with the initial point of $b$, i.e., $a(||a||) = b(0)$. The product $a.b$ of the paths $a$ and $b$ is defined by the formula: $$ (a.b)(t) = \left\{ \begin{array}{ll} a(t) & { 0 \leq t \leq ||a||}, \\ b(t - ||a||), & {||a|| \leq t \leq ||a|| + ||b||.} \end{array} \right. $$ It is obvious that this defines a continuous function.
My questions are:
1-Why we are sure that this defines a continuous function?
2- Is there an intuition behind this definition?
Could anyone help me answering these questions please?
1) As both of them are continuous and coincide on their interection.
2) Yes, let me show you a picture I once drew for that:
This drawing actually shows loops (paths with same start and termianl point) but the intuition is the same. You walk through the red loop $a$ and then through the blue loop $b$ to get the purple loop. So the multiplication is the concatenation.