Let $S^1$ be the unit circle in $\Bbb R^2$, with the subspace topology. Let $X \subset \Bbb R^3$ be given by $S^1 × [0, 1]$, and $Y \subset \Bbb R^2$ be $ \{(x, y)|1 \leq x^2 + y^2 \leq 2\}$. Show that $X$ and $Y$ are homeomorphic.
I know that to show that they are homeomoprhic, I must find a function from $X$ to $Y$ which is bijective and continuous and one inverse function that is continuous, but I don't know where to start the proof or find a function. How should I approach the problem?
Edit: Okay, so this is how they are look, however I feel like there's a problem because the empty part of the ring. I can't just make z=0 and map cylinder's insides to start from the circle with the radius 1 or can I?
The set $X$ is a cylinder and the set $Y$ is an annulus. Simply flatten the cylinder:$$\begin{array}{ccc}X&\longrightarrow&Y\\(x,y,z)&\mapsto&(z+1)\cdot(x,y)=((z+1)x, (z+1)y)\end{array}$$