TITLE: What is the difference between the sphere and projective space?
QUESTION [5 upvotes]: I know about the antipodal mapping.  
What I want to know is what the most significant differences between the sphere and projective space are, and how to think of each of them and their relationship to one another.
I come at this from a coding theory/vector quantization perspective; I'm trying to understand the difference between quantization techniques that codes a set of vectors in $R^n$ based on their direction vs. based on their oriented direction.

REPLY [3 votes]: On the sphere, any two "lines" intersect twice; in the projective plane, just once.
The sphere is oriented; the projective plane is not.  The letter R has more asymmetry than other letters of the alphabet; move and "R" around the projective plane and keep it going in the same direction until it returns to where it started.  It will be a backwards, mirror-image "R".
The sphere is simply connected: Draw any curve that returns to its starting point,  and it can be contracted to a point.  A line in the projective plane cannot be so contracted.