A surface, or 2-manifold, is a shape any small enough neighborhood of which is topologically equivalent to a neighborhood of a point in the plane. For instance, a the surface of a cube is a surface topologically equivalent to the surface of a sphere. On the other hand, if we put an extra wall inside a cube dividing it into two rooms, we no longer have a surface, because there are points at which three sheets come together. No small neighborhood of those points is topologically equivalent to a small neighborhood in the plane.
Figure 7: Here are some pictures of surfaces. The pictures are intended to indicate things like doughnuts and pretzels rather than flat strips of paper. Can you identify these surfaces, topologically? Which ones are topologically the same intrinsically, and which extrinsically?
Recall that you get a torus by identifying the sides of a rectangle as
in Figure 2.10 of SS (The Shape of Space).
If you identify the sides slightly differently,
as in Figure 4.3, you get a surface called a Klein bottle,
shown in Figure 4.9.