Why are spheres with two holes and cylinders homeomorphic? I started reading a book on topology and encountered the following in the preliminaries:
A sphere with two holes, a cylinder, an annulus, and a disc with one hole
are homeomorphic. A sphere with two holes is just an inflated version of a
cylinder, which flattens into an annulus (a disc with one hole).
Simply put I don't understand how I can inflate a cylinder into a sphere with two holes.
visualization from the book

$|\;\;|$  cylinder (understood as having no "cap" nor "base"; surely that's your issue)
$(\; \;)$ sphere with two holes.
Note that the surface around a cube (drawn by the faces) is homeomorphic to the (empty) cylinder with a cap and base, and it's homeomorphic to the sphere (without holes).
