TITLE: What do we lose in Projective Spaces?
QUESTION [5 upvotes]: We can think of the Complex Numbers as an extension of the Real Numbers, similarly we can think of the Projective Plane naturally as a nice extension of the Euclidean Plane. But, when we go from real to complex numbers we lose some structure, for example total ordering, we $\textit{can't}$ ask about ordering anymore.
$\textbf{Question:}$ What do we lose by working in the Projective Plane? In other words, what "euclidean" questions don't make sense in the projective context?

REPLY [12 votes]: You lose the vector space structure and even the affine structure and even the additive group structure when you go from $\mathbb C$ to $\mathbb P^1_\mathbb C$.
 More concretely,  given two points in  $\mathbb P^1_\mathbb C$ it doesn't make sense to talk of their difference as a vector.
The general context  is that $\mathbb C$ has both algebraic and geometric features while $\mathbb  P^1_\mathbb C$ has no algebraic structure whatsoever but has many geometric structures:    
As a topological space, manifold, Riemannian manifold, Riemann surface [ not  synonymous at all with the former !], conformal space, algebraic curve,...