This book grew out of courses given at Swansea University to second- and thirdyear
undergraduates. It is designed to provide enough material for a one-year
course and splits naturally into a preliminary topology course (Chapters 2–6)
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 textbooks,
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 oneyear
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 metric
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 introduction
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.