Surfaces

Most of the above examples are surfaces.

Definition
A surface is something in which every point has a region around it which is homeomorphic to a disc in R².

We will be most interested in what are called compact surfaces. These are closed (in the sense that they contain the limits of all sequences in them) and bounded subsets of Rⁿ.

Examples
See the above examples: a Sphere S², a Torus T, an (open) cylinder (which does not contain the upper and lower boundary circles and so is not compact), an (open) Möbius band (with the boundary removed -- so it is not compact either) and the Real Projective Plane.

The see how each point of the Projective Plane has the "right kind" of region around it, piece together the area around a boundary point B from "two bits".

More examples

1. The two-holed torus

   More generally, one can construct an n-holed torus or "surface of genus n".

2. A Klein bottle
   This is made by identifying the ends of a cylinder "the wrong way round" (so as not to get a torus!).
   We can describe this by starting with a square and identifying two edges to make the cylinder which we represent as:
   Then identifying the ends of the cylinder we get either a torus or a Klein bottle.

   Making the Klein bottle corresponds to identifying the edges of the square as:   

3. The Projective plane
    
   As we saw earlier this corresponds to identifying the opposite points on the boundary of a disc