A Mathematics of Topological Science

The word topology is derived from two Greek words, topos meaning surface and logs meaning discourse or study. Topology thus literally means the study of surfaces or the science of position. The subject of topology can now be defined as the study of all topological properties of topological spaces. A topological property is a property which if possessed by a topological space X, is also possessed by every homeomorphic image of X. If very roughly, we think of a topological space as a general type of geometric configuration, say, a diagram drawn on a sheet of rubber, then a homemorphism may be thought of as any deformation of this diagram (by stretching bending etc.) which does not tear the sheet. A circle can be deformed in this way into an ellipse, a triangle, or a square but not into a figure eight, a horse shoe or a single point. Thus a topological property would then be any property of the diagram which is invariant under (or unchanged by) such a deformation. Distances, angles and the like are not topological properties because they can be altered by suitable non-tearing deformations. Due to these reasons, topology is often described to non-mathematicians as “rubber sheet geometry”. Maurice Frechet (1878-1973) was the first to extend topological considerations beyond Euclidean spaces. He introduced metric spaces in 1906 in a context that permitted one to consider abstract objects and not just real numbers or n-tupls of real numbers. Topology emerged as a coherent discipline in 1914 when Felix Hausdorff (1868-1942) published his classic treatise Grundzuge der Mengenlehre. Hausdorff defined a topological space in terms of neighbourhoods of member sof a set.