Hopf–Rinow theorem

Hopf–Rinow theorem is a mathematical principle that explains how far two points can be apart to still be considered close. Imagine two points on a flat surface, like a sheet of paper. If those two points are close enough together, then they are close enough that, if you try to draw a line between them, the line will not have to curve. But if the points are far enough apart, then the line will need to curve around other points on the paper in order to connect them. Hopf–Rinow theorem tells us how far apart two points can be, and still be considered close.