B - Notes on topology, dimensions, measures, embeddings and homotopy

Summary

Topology had its origins in what some people would now call mathematical recreations. An early example was the problem of the seven bridges of Königsberg, solved by Euler in 1736, which involved showing that it is impossible to transverse each of seven bridges connecting two islands in a river to the banks exactly once. A much more profound problem, apparently first posed by Francis Guthrie to his teacher Augustus de Morgan (1852), is to prove that four colors suffice to distinguish the countries on any possible map (see, e.g., K.O. May, ‘The origin of the four-color conjecture’, Isis. 56, 346–8 (1965). It might be noted that the controversial concept of a ‘computer assisted proof’ of this conjecture has been employed by Appel and Haken (1977, 1986), using over 1000 hours of computer time). The term ‘topology’ was first introduced by J.B. Listing in 1847, and was roughly associated with the analysis of placements, and hence was frequently referred to as ‘analysis situs’ by Poincaré and others. Poincaré however is generally credited with being responsible for raising this method of analysis to the level of a branch of mathematics. Topology is now a diverse field of research, with a number of specialized branches. Our needs in topology are very modest, and the present discussion is largely limited to these needs. Fortunately there exists a number of good introductions to topology, some of which are listed in the references, which the reader should consult to augment this meager introduction.