Continuing with this post:
Milnor's appendix on classification of 1-manifolds
I want to show that $h$ is well defined. For that I must prove:
$\theta:=\dfrac{2\pi t}{\alpha-a}\in<0,2\pi>$.
Knowing that $t\in <a,\beta>$ I Just get to
$$a<t<\beta\to 0<t-a<\beta-a$$
$$0<\dfrac{1}{\beta-a}<\dfrac{1}{t-a}$$
To appear $\alpha-a$, do I have to analyse many cases? Or is there a simpler way? Then have to analyse in the overlaps.
Thank you.