Topology and Homotopy

Abstract

The best way to understand the notion of topology is perhaps to use the viewpoint of Felix Klein’s Erlangen Programme. According to this programme, various branches of abstract mathematics can be classified by means of the groups of transformations which preserve the properties of objects studied in that branch. For example, Euclidean geometry can be considered to be the study of the properties of objects which are left invariant under the Euclidean group of motions, i.e., rigid rotations and rigid translations. (For some of the important theorems of Euclidean geometry, the ones dealing with similar triangles, it may be necessary to also include uniform scalings.) Projective geometry, on the other hand, can be considered to be the study of properties left invariant under projective transformations. In this sense, topology is just the study of properties left invariant under homeomorphisms, i.e., bicontinuous mappings.