In recent years, many students have been introduced to topology in high
school mathematics. Having met the Mobius band, the seven bridges of
Konigsberg, Euler's polyhedron formula, and knots, the student is led to
expect that these picturesque ideas will come to full flower in university
topology courses. What a disappointment "undergraduate topology" proves
to be! In most institutions it is either a service course for analysts, on abstract
spaces, or else an introduction to homological algebra in which the only
geometric activity is the completion of commutative diagrams. Pictures are
kept to a minimum, and at the end. the student still does n<-t understand the
simplest topological facts, such as the reason why knots exist