Suppose that $D$ is a plane domain. Let $N$ be the set of all integers $n$ such that there is a closed, piecewise smooth curve $\gamma$ in $D$ whose winding number about the origin equals $0$: $$N:=\{n\in\mathbb{Z}\mid \textrm{there is a closed, piecewise smooth curve }\gamma \textrm{ in } D \textrm{ such that } n(\gamma ,0)=n\}.$$

(Here winding number $n(\gamma ,0)$ of a closed, piecewise smooth ($C^1)$ curve $\gamma:[a,b]\rightarrow \mathbb{C}$ is defined by $n(\gamma,0):=\int_\gamma \frac{1}{z}dz.$)

I was thinking about what the set $N$ look like in general. First I noticed that $N$ must be the set of the form $k\mathbb{Z}$, where $k$ is a nonnegative integer. After some drawing, I started to suspect that $N$ is either $\{0\}$ or $\mathbb{Z}.$ However, I can neither prove nor disprove it. Is it true that $N=\{0\}$ or $\mathbb{Z}?$ If so, why?

Any help is appreciated. Thanks in advance!


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