In this chapter we study the differential geometry and topology of the projective plane. We start by a short exposition on the classification of surfaces and the notions of compactness and orientability. We show that the projective plane is a compact, nonorientable surface. We also show that the projective plane can be embedded in four-dimensional space. We state without proof a theorem of Banchoff that says that any immersion of the projective plane in three-dimensional space must have a point where the surface intersects itself at least three times. We also show computer-generated images of Boy's surface, which show that an immersion with a single triple point exists.
In the last part of this chapter we discuss some theorems in projective topology such as Moebius theorem and a recent theorem of Ghys on the zeros of the Schwartzian derivative.
On the whole, this chapter differs from the others in that many details are skipped and the student is not expected to understand 100% of the material. The aim is to motivate and foreshadow future courses on differential geometry and topology.