Believe it or not, we have almost all the topological ingredients for making any surface whatsoever. Only one very important surface remains to be explored and of course we need a way to put surfaces together to make new surfaces. First thing first.
Let's abbreviate green forward and orange backward as purple forward (see illustration). This is another common way to define the projective plane. Topologically it's equivalent to the first—just rename the arrows and smooth the two sharp corners in the fundamental polygon. This is again topologically equivalent to taking a hemisphere (half a sphere) and identifying points on the boundary with their antipodes. It is this form, with the inherent spherical geometry that defines the geometry of the projective plane.
Before we go on any further let's summarize and make sense of what we've learned so far. A surface is nonorientable if you can walk along some path and come back to where you started but reflected, as on a Möbius band. In fact a surface is nonorientable if and only if you can find a Möbius band inside of it, like we did in the Klein bottle and the projective plane. A surface is orientable if it's not nonorientable: you can't get reflected by walking around in it. Two surfaces are topologically equivalent if we can deform one into the other without tearing and geometrically equivalent if your avatar the cyclops can't tell the difference between them by looking around. Two surfaces which are geometrically equivalent are indeed equivalent topologically, but not the other way around. Here's a pictionary of the surfaces we've seen so far (except the Möbius band, which, having a boundary, is the ugly duckling of the bunch and is thus omitted).
Until the end of this section, we'll deal solely with topological properties of surfaces and ignore any questions of geometry. Given any two surfaces, there's a very useful way of putting them together, called a connected sum and usually denoted by #. Suppose S and T are two surfaces. Then we get their connected sum S#T as follows.
There are two very important theorems about surfaces that'll be of
interest to us, one concerning surfaces as topological object and one
concerning them as geometric object. The first of those is this:
A closed surfaces is simply one that's finite in extent. A plane is not a closed surface for example, but a sphere is. Also note that this only applies to surfaces without boundaries, thus the Möbius band, for instance is not listed. By the previous activity, all the surfaces on the left and the sphere are orientable, while all the surfaces on the right are nonorientable.
From the last activity we see that the operation # doesn't have inverses. Even though P#P#P is topologically equivalent to (Torus)#P, where P is the projective plane, clearly P#P=Klein bottle is not topologically equivalent to the torus, so we can't "cancel a P on both sides of the equation".Next: Geometries of Surfaces