Show that $\Theta=S^1\cup (\{0\}\times [-1,1])$ is homotopy equivalent to $\infty=(S^1+(1,0))\cup (S^1+(-1,0))$ where $(S^1 + (a, b))$ is the circle described by $(x - a)^2 + (y - b)^2 = 1$.
I can intuitively see that both are deformation retracts of $\mathbb R^2\setminus\{p,q\}$. However, I am interested in finding explicit maps for these deformation retractions.