This question is more to satiate curiosity. As an example, R^2 as a set only has no geometrical properties to it. But adding the Euclidean metric to the set creates a metric space. Further than that though, the metric also allows us to impose R^2 onto a plane to gauge the distance between two points visually. I was wondering if there was other metrics of R^2 that would create for example a hyperbolic paraboloid or a sphere (maybe with one point removed)?