This is a book organized into five main chapters of “appealing….ideas and problems involving curves surfaces and knots” together with an introductory chapter in which ‘rubber sheet’ geometry is reviewed and the process of increasing the formality and precision of intuitive ideas is begun. This additional precision involves the compactness condition for the surfaces to be studied (surface includes its boundary) the definition of continuous deformation (bicontinuous bijection) and the Euclidean metric for Rn (for n=1 2 3 and sometimes 4). Also introduced in this chapter is the idea of plane diagrams of cylinder torus Mobius strip with their associated edge and vertex labelling. The diagrams are attractive and helpful and should encourage students to draw their own or even construct some of the surfaces. Further reinforcement of the topological differences between these surfaces comes from the idea of playing tic-tac-toe (noughts and crosses to us) on the surface of a torus or Klein bottle
