a topologist would say they have different fundamental groups
In the affine real plane, yes. In the projective real plane, all (non-degenerate) conics have the same fundamental group.
In the affine real plane, yes. In the projective real plane, all (non-degenerate) conics have the same fundamental group.
On a projective plane a parabola contains a single point at infinity, connected (both figuratively and topologically) to the two open ends. A hyperbola contains two distinct points at infinity each connected to one end of each of the two usual components.