  A simple, yet important, example of a Lie group is the group SO(2), instantiated by the rotations of the plane about an arbitrary fixed point. The manifold is topologically compact and therefore cannot be covered by a single coordinate patch, but three will suffice: one including, say, all counterclockwise rotations by more than three radians and less than four, which can be naturally coordinatized using the real numbers in the open interval (3,4); another patch comprising the inverses of the former, which can be mapped onto the open interval (−4,−3), and a third one covering all counterclockwise rotations by less than two right angles plus their clockwise inverses, which can be mapped onto the open interval (−π,π). Indeed, all the groups we encountered in §3, which Klein used for characterizing the Euclidean geometry of space and the classical Non-Euclidean geometries, are Lie groups, and their respective smooth manifold structures allow for topological quirks. Thus the Euclidean isometries constitute a disconnected manifold, with mirror reflection not included in the same component as the subgroup of Euclidean motions. 