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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 **R**^{2}.

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 **R**^{n}.

**Examples**

See the above examples: a Sphere *S*^{2}, 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**

**The two-holed torus**More generally, one can construct an

*n*-holed torus or "surface of genus*n*".**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:

**The Projective plane**

As we saw earlier this corresponds to identifying the opposite points on the boundary of a disc

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