I am trying to prove a proposition relating analysis and geometry. I have a general idea on how to prove it. However, a small part of the proof needs a lemma about path homotopy and winding number. Specifically, here it is:
Let $\alpha,\beta:[0,1]\rightarrow\mathbb{C}\setminus\{p\}$ be two (continuous) paths (not necessarily closed) such that $\alpha(0)=\beta(0)$ and $\alpha(1)=\beta(1)$.
If $\alpha\simeq_\mathrm{p}\beta$, then $\mathrm{W}(\alpha,p)=\mathrm{W}(\beta,p)$.
I tried to find this theorem in books, but so far, I only found the same lemma but applied to closed curves. I found someone stating this here, but with no proof. In fact, he/she said that it can be proved easily by lifting lemma. However, I didn't take Algebraic Topology course yet for now. I have not much idea about the lifting lemma.
I really appreciate if someone could provide a proper explanation / proof / idea / reference for this lemma. Thank you in advance.