In 1872, Felix Klein made a stunning application of groups to geometry, which introduced a beautiful order into the then existing chaos of geometrical information. He defined geometry as follows: A geometry is the study of those properties of a set S that remain invariant when the elements of the set S are subjected to the transformations belonging to some transformation group G (a transformation group, is a group consisting of invertible functions on S, the operation being their composition). Thus in one stroke, Klein developed the idea through which the generalized concept of a geometry could be understood through a group. If one wants to create a new geometry, one has to choose three things - the space or manifold, i.e. the class from which the elements from S have to be picked (e.g. plane, spherical surface etc), the fundamental elements of the geometry (e.g. point, line etc) and the group of transformations to which the fundamental elements are to be subjected. The construction of a new geometry, in this way, is a rather simple matter.
Let us consider three commonly known geometries as examples. The first is the plane Euclidean metric geometry commonly taught in schools. Consider the Euclidean plane, R^2. The fundamental elements are points for us. So let S=R^2. Consider the totality of the following three invertible functions acting on S:
1. Rotation (here a point (x,y) is mapped to (x cos A-x sin A, y sin A + y cos A) where A belonging to {-pi}U(-pi,pi) is the angle through which we rotate (x,y).
2. Translation (here a point (x,y) is mapped to (x+A,y+B)).
3. Reflection about the x axis (here the point (x,y) is mapped to its conjugate i.e. (x,-y)).
It can be easily shown that the collection of all functions which are of the above three types generates a group M which amazingly contains all possible isometries (or distance preserving maps) possible in the plane. This group M is called the group of rigid motions. Thus any map in M maps a triangle to another congruent triangle, which other then the fact that it may be tilted or further off, is otherwise exactly the same as the previous one. Thus the motion of this triangle as it takes off from its previous position to the current one, is entirely rigid. There is no contraction or deformation involved in the course of the motion. Hence we derive the name rigid motion. Just like the property of being congruent is preserved be these motions, so too are the properties of parallelism, perpendicularity, similarity of figures, collinearity of points and concurrence of lines. This is the geometry within which we worked in our school when we examined these properties and their implications in detail.
Let us now enlarge M. Consider the totality of all the transformations which act as follows on (x,y). Let x be transformed to (ax+by+c)/(gx+hy+i) and y to (dx+ey+f)/(gx+hy+i) where a,b,c.. are real numbers such that the determinant a b c, d e f, g h i is non-zero. (Here a b c is the first row, d e f the second and g h i the third). The geometry thus obtained is called plane projective geometry. Under this new geometry of the afore mentioned properties only collinearity of points and concurrence of lines remains invariant, and perpendicularity etc cease to remain so. An important invariant (which was also obviously invariant earlier) is the cross ratio of four collinear points.
The final geometry which we shall examine is a further generalization of projective geometry. Consider the totality of all transformations which have the following form: transform x to f(x,y) and y to g(x,y) where f(x,y) and g(x,y) are continuous and single valued functions with continuous inverses (or are homeomorphic). This recently developed branch of geometry is known as topology. Topology of the plane is often referred to as "rubber sheet" geometry, for in stretching or contracting a rubber sheet, the points of the sheet undergo just such a bicontinuous single valued transformation.
Topology is the deepest geometry in the sense considered above, its transformation group embraces both the groups of rigid motions and of projective geometry. So any theorem of topology holds within these other geometries. (The converse is false obviously). The invariant properties such as a simple closed curve remain simply closed after transforming, and deletion of a single point doesn't disconnect the curve etc, are referred to as topological properties.