Let $X$ be the space obtained from the sphere $S^2$ by gluing the north pole to the south pole, let $Y=\mathbb{R}^3-S^1$, where $S^1=\left\{(x,y,0)\in\mathbb{R}^3:x^2+y^2=1\right\}$ and let $Z$ be the union of a torus of revolution with a disk whose boundary is the smallest of the parallels of the torus. Prove that $X$,$Y$ and $Z$ have the same homotopy type.
How can I prove that $Z$ have the same homotopy type with to $X$ or $Y$?
Goal
Provide a plan along with some guidance for the reader to complete the details.
Notation: $\sigma=\sqrt{x^2+y^2}$. $Y=\mathbb R^3\setminus\{\sigma=1,\;z=0\}$.
Plan
Step 1. Define $B_2=\{(x,y,z)\mid x^2+y^2+z^2\le4\}$. Replace $Y$ with $\bar Y=B_2\setminus\{\sigma=1,\;z=0\}$.
Step 2. Define $(0,0,z)\sim(0,0,0)$. Replace $\bar Y$ with $\bar Y/{\sim}$.
Step 3. Replace $\bar Y/{\sim}$ with $T^*=\{(\sigma-1)^2+z^2\le1\}\setminus\{\sigma=1,\;z=0\}$.
Step 4. Replace $T^*$ with $T=\{(\sigma-1)^2+z^2=1\}$.
Guidance
Step 1. Put $\rho=\sqrt{\sigma^2+z^2}$. $$ f(x,y,z)=\begin{cases} \tfrac{2}{\rho}(x,y,z) &\rho\ge 2,\\ (x,y,z) &\rm otherwise. \end{cases} $$
Step 2. Define $g\colon\bar Y/{\sim}\to\bar Y$ as $g\colon[x,y,z]\mapsto(x,y,\frac{\sigma}{2}z)$.
Step 3. Write $\zeta=\sqrt{1-(\sigma-1)^2}$. Define $h\colon\bar Y/{\sim}\to T^*$ as $$ h([x,y,z]) = \begin{cases} (x,y,\zeta) &z\ge\zeta,\\ (x,y,z) &-\zeta\le z\le\zeta,\\ (x,y,-\zeta) &z\le-\zeta. \end{cases} $$
Step 4. Write $\xi=\sqrt{(\sigma-1)^2+z^2}$. Define $r\colon T^*\to T$ as $$ r(x,y,z) = \begin{cases} \tfrac{1}{\xi}(x-x/\sigma,y-y/\sigma,z)+(x/\sigma,y/\sigma,0) &\sigma\ne0,\\ (0,0,0) &\sigma=0. \end{cases} $$
To do