Finding a bijection and inverse to show there is a homeomorphism I need to find a bijection and inverse of the following:
$X = \{ (x,y) \in \mathbb{R}^2 : 1 \leq x^2 + y^2 \leq 4 \}$ with its subspace topology in $\mathbb{R}^2$ 
$Y = \{ (x,y,z) \in \mathbb{R}^3 : x^2 + y^2 = 1$ and $ 0 \leq z \leq 1 \}$ with its subspace topology in $\mathbb{R}^3$
Then show they are homeomorphic. Not sure where to start.
 A: Hint: $X$ is an annulus, and $Y$ is a cylinder. Imagine "flattening" a cylinder by forcing its bottom edge inwards and its top edge outwards; it will eventually become an annulus. This is the idea of the map; you should work out the actual expressions describing it on your own.

Here is a gif animation I made with Mathematica to illustrate the idea:


F[R_][t_, x_, y_] := {(R + x) Cos[t], (R + x) Sin[t], y}

BentCylinder[R_, r_, s_, t_, z_] := F[R][t, r + s*Sin[z], s*Cos[z]]

BendingCylinder[R_, r_, z_] := 
 ParametricPlot3D[
  BentCylinder[R, r, s, t, z], {s, -r, r}, {t, 0, 2 Pi}, Mesh -> None,
   Boxed -> False, Axes -> None, PlotStyle -> Red, 
  PlotRange -> {{-10, 10}, {-10, 10}, {-5, 5}}, PlotPoints -> 50]

Export["animation.gif", 
 Table[BendingCylinder[6, 2, z], {z, 0, Pi/2, 0.02*Pi}], 
 "DisplayDurations" -> {0.25}]

A: You can use the similar idea of Stereographic Projection. Take a point above the cylinder along the axix, say $P:=(0,0,2)$. Now join $P$ by a straight line to each point of the cylinder say $p$  and extend it till it hits the $x-y$ plane at point, say $p^\prime$. Then the map $p\mapsto p^\prime$ maps the cylinder to an annulus. Now scale that annulus properly to get your desired annulus.
