If one only considers the Euclidean topology of the plane and the topology inherited by "Q", then the lines bounding "Q" seem close to "Q". Insight from the metric space "HP" shows that the open set "Q" has only the origin as boundary when viewed through the correspondence. Indeed, consider rays from the origin in "Q", and their images, vertical rays from the boundary "R" of "HP". Any point in "HP" is an infinite distance from the point "p" at the foot of the perpendicular to "R", but a sequence of points on this perpendicular may tend in the direction of "p". The corresponding sequence in "Q" tends along a ray toward the origin. The old Euclidean boundary of "Q" is no longer relevant.