## Torus

Let , the product of two circles, and the surface of revolution formed by rotation of the circle in the plane around the –axis:

And last, consider in the unit square the equivalence relation defined by if one of the following is satisfied:

- and , or
- and or
- and

Let be the quotient set defined through equivalence classes from the relation above.

We want to prove that these three spaces , and are actually homeomorphic. In order to construct the corresponding homeomorphisms we will use the following intuitive ideas, as illustrated in the images of the sets below:

For example, to find a homeomorphism , we consider a construction based on the following facts:

- The horizontal plane intersects the torus in a single circle (represented in pink). We identify the first copy of in with this circle. Let us denote that pink circle
- Once chosen an element in the first copy of from , we consider the corresponding point in from a suitable homeomorphism between and Let us write
- Consider the intersection of the torus with a plane that goes through the origin, the point , and that is vertical. This intersections consists on two different circles, only one of which goes through the point (represented in red); let us denote this new circle, which we want to identify with the second copy of in
- Once chosen an element in the second copy of from consider the corresponding point in from another suitable homeomorphism. We write

I leave to the reader the pleasure of finding an analytic expression for this homeomorphism

To construct an homeomorphism , we can proceed intuitively from a square piece of elastic material, and glue two parallel sides together as indicated in the diagram, thus forming a cylinder. It should not be too hard to construct a homeomorphism that turns a square into a cylinder in the Euclidean 3–dimensional space (use the correct coordinate system!) Notice that the bases of such cylinder (which correspond to the other parallel sides of the previous square) turned into circles. We can then glue both circles together as indicated in the diagram, thus forming a surface similar to that of This operation has also a nice analytic expression as a map from the cylinder to the torus, which once chosen the right coordinate system, should be no trouble to compute for the reader. The composition of these two homeomorphisms is, of course, an homeomorphism

Of course, to come up with the homeomorphism between and we can make use of the two previous: for example.