projective line

The projective line (over a field k) denoted P¹(k) is defined in the writetup on projective space. Let us describe what the real projective line and complex projective line look like topologically.

The real projective line P¹(R) consists of all [a,b], where a,b are real numbers not both zero, and we have that two such points are equal if and only if they lie on the same line in R². So we can choose a representative point for each of these lines on the unit circle. After we have done that we can see that the real projective line is the unit circle with the points x and -x identified. Each such point is represented by a point in the upper semicircle. Finally the extreme left hand point of the semicircle (-1,0) and the extreme right hand point (1,0) are identified, so again we obtain a circle. Thus topologically P¹(R) is a circle.

The complex projective line is quite different. Think about the sphere in real 3-space centred at (0,1,0) with radius 1. It is sitting on the plane z=0. (It's helpful to draw a picture of this). We can use stereographic projection to map the 2-sphere down onto the z=0 plane. It goes like this. For each point p on the sphere except for its north pole (0,2,0) we map it down onto the plane by considering the line through the north pole and p and choosing the point where it hits the plane. This gives a bijection from the points of the sphere without the north pole and the points of the plane. Thus the sphere consists of the points of the complex plane C together with an extra point at infinity. Topologically P¹(C) is the Riemann sphere S².