# 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

• If you held one end of the cylinder shut and blew into the other end, the mantle would inflate to become a sphere with two holes. Commented Mar 30, 2021 at 19:59
• This isn't enough material for a proper answer, but perhaps you are including the circle on top and bottom of the cylinder in your mental image and the author of this book is not including those since the formal definition of the cylinder is: ... The surface formed by the points at a fixed distance from a given straight line called the axis of the cylinder. Commented Mar 30, 2021 at 20:05
• Are you thinking about a ball, which includes the interior, as opposed to a sphere, which is just the surface? Think of taking the surface of the earth and poking two holes in it. It looks much like a cylinder, which has a hole at each end. Commented Mar 30, 2021 at 20:13

$$|\;\;|$$ 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).

• ASCII graphics win again. Commented Mar 30, 2021 at 20:14
• epic visualisation Commented Mar 30, 2021 at 20:14

The ASCII graphics look great!

Nonetheless, I recently made a mini animation in Mathematica, so I'm adding them here as an answer, in case someone finds them useful :)

You should think about the stereographic projections giving you a diffeomorphism from the sphere with a point removed to the plane. To be precise, you are projecting an emisphere to the plane, we can imagine the upper one. Now, you can take a spehere, remove two points (or two holes, doesn't really make a difference) and imagine this sphere to be contained in a cylinder. The same trick from both the emispheres gives you the homeomorphism you were searching for.

This is just an euristic consideration, I invite you to take an introductury text about differential geometry and you will understand more deeply.