This book grew out of courses given at the University of Wales Swansea to second- and third-year undergraduates. It is designed to provide enough mate- rial for a one-year course and splits naturally into a preliminary topology course (Chapters 1–6) and a follow-on course in algebraic topology (Chapters 7–11).
It is often said that topology is a subject which is poorly served for text- books, and when preparing the lecture courses I found no book that was both accessible to our undergraduates and relevant to current research in the field. This book is an attempt to fill that gap. It is generally accepted that a one- year course on topology is not long enough to take a student to a level where she or he can begin to do research, but I have tried to achieve that as nearly as possible. By omitting some of the more traditional material such as met- ric spaces, this book takes a student from a discussion of continuity, through a study of some topological properties and constructions, to homotopy and homotopy groups, to simplicial and singular homology and finally to an in- troduction to fibre bundles with a view towards K-theory. These are subjects which are essential for research in algebraic topology, and desirable for students pursuing research in any branch of mathematics. In fact, if I may be so bold as to say so, the subjects covered by this book are those areas of topology which all mathematics undergraduates should ideally see. In that sense, the material is essential topology.