= Synthetic geometry of the real projective plane
It good to think about how <Euclid's postulates> look like in the real projective plane:
* two parallel lines on the plane meet at a point on the sphere!
Since there is one point of infinity for each direction, there is one such point for every direction the two parallel lines might be at. The <parallel postulate> does not hold, and is replaced with a simpler more elegant version: every two lines meet at exactly one point.
One thing to note however is that ther <real projective plane> does not have <angles> defined on it by definition. Those can be defined, forming <elliptic geometry> through the <projective model of elliptic geometry>, but we can interpret the "parallel lines" as "two lines that meet at a point at infinity"
* points in the real projective plane are lines in <\R^3>
* lines in the real projective plane are planes in <\R^3>.
For every two projective points there is a single projective line that passes through them.
Since it is a plane in <\R^3>, it always intersects the real plane at a line.
Note however that not all lines in the real plane correspond to a projective line: only lines tangent to a circle at zero do.
Unlike the <real projective line> which is <homotopic> to the <circle>, the <real projective plane> is not <homotopic> to the <sphere>.
The <topological> difference bewteen the <sphere> and the <real projective space> is that for the <sphere> all those points in the x-y circle are identified to a single point.
One more generalized argument of this is the <classification of closed surfaces>, in which the <real projective plane> is a <sphere> with a hole cut and one <Möbius strip> glued in.