Geometry and Topology
A collection G of transformations constitutes a group if
1. The composition of arbitrary mappings in G is itself in G;
2. The identity transformation is in G;
3. Every transformation in G is invertible and the inverse is in G.
As formulated by the 19th century mathematician Felix Klein, a geometry comprises properties preserved by a transformation group. The set upon which elements of G act is the "universe" of the geometry. The geometry is not associated to the universe by itself, but to the transformation group G.
In this viewpoint, "Euclidean (plane) geometry" is the study of rigid plane motions, transformations preserving length and angle. The universe is the plane, but the geometry itself is expressed by special subsets of the plane, such as lines, rays, segments, and circles.
Similarly, "topology" is the study of properties preserved by continuous mappings with continuous inverse. For subsets of the plane, such a mapping may be loosely visualized as the act of stretching a sheet of rubber without tearing or gluing. For this reason, mathematicians sometimes use the phrase "rubber sheet geometry" to connote topology.
In this image collection, "topology" refers more specially to mappings between surfaces that are differentiable with differentiable inverse. "Geometry" refers to surfaces equipped with concepts of length and angle, and smooth mappings that preserve length and angle.