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.

- To the right is the promised projective plane, defined as usual by identification of a square's edges. Is it flat like the torus and Klein bottle, or does it have cone points?
- Can you find a Möbius band in this surface? How about another one?

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.

- As a brave adventurer, you've made your way onto the projective plane. As you look around you, what do you see? (Hint: remember the analysis of this question we did for the sphere. Use the same reasoning to think about how light rays travel on the projective plane, and do draw pictures.)
- If you want to see the top of your head, would you look "up" or "down"?
- You watch a friend walk away from you along a great circle until he meets you having walked "all the way around". What does his journey look like?
- What's different about him when he returns?

This definition of topological equivalence is
slightly oversimplified. The general notion involves the concept of
*homeomorphism*, a map between surfaces which respects their
topological structure, and can allow some restricted tearing and
regluing.

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.

- Cut out a disk from S and from T.
- Glue the boundary of the cut out disk in S to the boundary of the cut out disk in T. This is all done topologically, so we can stretch and shrink freely.

- Suppose C is some surface, and S is the sphere. What is C#S? (Hint: try it with C being one of the surfaces you're familiar with.)
- What surface is the projective plane with a disk removed? (Hint: do some cutting/pasting and deforming. If you're having a hard time with this admittedly not so simple problem, see this illustration.)
- Let P be the projective plane. What familiar surface is P#P? (Hint: Look back to problem 3 in the last section.)
- Given any surface K and a nonorientable surface C, can the surface K#C be orientable? (Hint: recall that nonorientable means that there's a "reflecting" closed path in the surface C. What if that path does not go though the disk we removed from C to glue it to K? What if it does?)
- Here's a more challenging problem. Show that if surfaces C and K are both orientable, then so is C#K. (Hint: if C#K is not orientable, then you can find a Möbius band inside it.)

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 surfaces listed in the theorem, which is the Klein bottle? How about (Klein bottle)#(Klein bottle)?
- Show that (Klein bottle)#(Möbius band) is topologically equivalent to (Torus)#(Möbius band). (Hint: this is not so simple and requires some playing around with the surfaces. Here's the first illustrated hint, and here's the second.)
- Which surface is (Klein bottle)#torus? (Hint: use part 2 above.)

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