The projective

line (over a field

*k*) denoted

**P**^{1}(

*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 **P**^{1}(**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
**R**^{2}. 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 **P**^{1}(**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 ****P**^{1}(**C**) is the Riemann sphere
*S*^{2}.