# Thread: complex numbers - winding number

1. ## complex numbers - winding number

Hi all,
I need to prove the following lemma in attachment . Our instructor wants us to do it in the following way:
Let A be the set of all $x \in [0,1]$ with the property that there exists a continuous function
$\theta : [0,x] -> R$ such that $\theta (0) = \theta_{0}$
and

$\gamma (t) = | \gamma (t) | e^{i \theta (t) } t \in [0,x]$

Prove that A is nonempty , closed and relatively open in [0,1] and conclude that A = [0,1] since [0,1] is connected.

Can anyone help?? I don't even have a slight idea how to proceed.

2. Notice you can bound $\gamma$ away from zero and this together with the unifrom continuity of said function allows you to pick a neighbourhood (in $[0,1]$ ) of any point in A on which there is a branch of logarithm defined. There may be some ambiguity over multiples of $2\pi$ but the exponential doesn't see them, so you can ignore them and take the "true" extension. This proves A is both open and closed, and A is never empty because 0 is always in your set. I leave the formal proof to you.

3. OK, I saw that the set A is open. How do we see it's closed? I can't see why the complement should be open.