This is a book of elementary geometric topology, in which geometry, frequently illustrated, guides calculation. The book starts with a wealth of examples, often subtle, of how to be mathematically certain whether two objects are the same from the point of view of topology. After introducing surfaces, such as the Klein bottle, the book explores the properties of polyhedra drawn on these surfaces. More refined tools are developed in a chapter on winding number, and an appendix gives a glimpse of knot theory. Moreover, in this revised edition, a new section gives a geometrical description of part of the Classification Theorem for surfaces. Several striking new pictures show how given a sphere with any number of ordinary handles and at least one Klein handle, all the ordinary handles can be converted into Klein handles. Numerous examples and exercises make this a useful textbook for a first undergraduate course in topology, providing a firm geometrical foundation for further study. For much of the book the prerequisites are slight, though, so anyone with curiosity and tenacity will be able to enjoy the Aperitif. "…distinguished by clear and wonderful exposition and laden with informal motivation, visual aids, cool (and beautifully rendered) pictures…This is a terrific book and I recommend it very highly." MAA Online "Aperitif conjures up exactly the right impression of this book. The high ratio of illustrations to text makes it a quick read and its engaging style and subject matter whet the tastebuds for a range of possible main courses." Mathematical Gazette "A Topological Aperitif provides a marvellous introduction to the subject, with many different tastes of ideas." Professor Sir Roger Penrose OM FRS, Mathematical Institute, Oxford, UK.