The following is a completely general definition of projective space.
Definition 1.1 (Projective Space)
Let be a field and an integer.
Then dimensional-projective space is, as a set,
not all $a_i=0$
where is the equivalence relation in which
for all nonzero .
(When has a topology, inherits a topology, which we
probably won't worry about anymore in this course.)
When , the projective space of dimension 0 is a single point.
When , the projective line is, as a set,
Thus the projective line is the usual line union one extra point
, which we often think of as being ``at infinity''.
The set
is the real line along with one extra point at
infinity; this is in bijection with a circle.
The set
is equal to the complex plane
along with one extra
point at infinity. Alternatively,
can be thought of as
the points on the sphere with the north pole corresponding to the
point at infinity.
When we obtain the projective plane:
We can think of as the usual plane along with
a copy of ``at infinity''. The real projective plane
looks like a plane union a circle at infinity.
The complex projective plane
has real dimension
so it is harder to describe, but it is where we will primarily work.
In general, is usual -dimension space along with
a
``at infinity''.