I know that if $\Omega$ is open subset of $\mathbb{C}$ and $\gamma_1$,$\gamma_2$ are two homotopic paths such that $\gamma^*_1$,$\gamma^*_2 \subset \Omega$ then $\forall \alpha$$\in$ $\mathbb{C}\setminus \Omega$ $\quad$$n(\gamma_1,\alpha)=n(\gamma_2,\alpha)$.
I also know that the winding number equals $0$ in the unbounded component of $\mathbb{C}\setminus\gamma^*_i$. But if $\gamma^*_i\subset \Omega$ why $n(\gamma_i,\alpha)=0$? Is not true that $\mathbb{C}\setminus\Omega \subset A$ where $A$ denotes de unbounded component of $\mathbb{C}\setminus \gamma^*_i$ ?
I'd be very thankful if someone clarified these concepts.