Three equivalent assertions in pg.14 in Ralph H. Fox book of knot theory.

Question

The book says:

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., $a(||a||) = b(0)$.It is obvious that the three assertions:

(i)$a.b$ and $b.c$ are defined.

(ii) $a.(b.c)$ is defined.

(iii) $(a.b) . c$ is defined.

are equivalent.

But I do not know how is this obvious, could anyone help me to prove them please?

Answer 1

$i)\Rightarrow ii)$ If $a\cdot b$ is defined then $b(0)=a(\|a\|)$. If $b\cdot c$ is defined then $(b\cdot c)(0)=b(0)=(\|a\|)$. Hence $a\cdot(b\cdot c)$ is defined.

$ii)\Rightarrow iii)$ If $a\cdot (b\cdot c)$ is defined then $a(\|a\|)=(b\cdot c)(0)=b(0)$ so $a\cdot b$ is defined. Also this implies that $(a\cdot b)(\|a\cdot b\|)=b(\|b\|)=c(0)$, so that $(a\cdot b)\cdot c$ is defined.

$iii)\Rightarrow i)$ If $(a\cdot b)\cdot c$ is defined then clearly $a\cdot b$ is defined. Moreoreover $(a\cdot b)(\|a\cdot b\|)=b(\|b\|)=c(0)$, so $b\cdot c$ is defined.