# Gluing two points of a annulus on the boundaries then its boundaries are not homotopic

Let $$A$$ be a annulus $$\{ (r,\theta) \vert 1\le r \le 2 , \theta \in [0,2\pi] \}$$. Gluing $$(1,0)$$ and $$(2,0)$$ together. Then how to prove that $$\{r=1 \}$$ and $$\{ r = 2 \}$$ are not homotopic rel $$(1,0) \thicksim (2,0)$$.

The difficulty is, for example, let $$h_t(s)$$ is the homotopy such that $$h_0$$ is the outer circle and $$h_1$$ is the inner circle. Then with $$t$$ and $$s$$ very small $$h_t(s)$$ may go through the point $$(1,0)$$ to the other side of the annulus. Just like the diagram below.