### Synopses & Reviews

Taking a direct route, Essential Topology brings the most important aspects of modern topology within reach of a second-year undergraduate student. Based on courses given at the University of Wales Swansea, it begins with a discussion of continuity and, by way of many examples, leads to the celebrated "Hairy Ball theorem" and on to homotopy and homology: the cornerstones of contemporary algebraic topology. While containing all the key results of basic topology, Essential Topology never allows itself to get mired in details. Instead, the focus throughout is on providing interesting examples that clarify the ideas and motivate the student, reflecting the fact that these are often the key examples behind current research. With chapters on: * continuity and topological spaces * deconstructionist topology * the Euler number * homotopy groups including the fundamental group * simplicial and singular homology, and * fibre bundles Essential Topology contains enough material for two semester-long courses, and offers a one-stop-shop for undergraduate-level topology, leaving students motivated for postgraduate study in the field, and well prepared for it.

#### Review

From the reviews: "This book presents the most important aspects of modern topology, essential subjects of research in algebraic topology ... . The book contains all the key results of basic topology and the focus throughout is on providing interesting examples that clarify the ideas and motivate the student. ... this book contains enough material for two-semester courses and offers interesting material for undergraduate-level topology, motivating students for post-graduate study in the field and giving them a solid foundation." (Corina Mohorianu, Zentralblatt MATH, Vol. 1079, 2006) "This text provides a concise and well-focused introduction to point set and algebraic topology. The main purpose is to quickly move to relevant notions from algebraic topology (homotopy and homology). Throughout the book the author has taken great care to explain topological concepts by well-chosen examples. It is written in a clear and pleasant style and can certainly be recommended as a basis for an introductory course on the subject." (M. Kunzinger, Monatshefte für Mathematik, Vol. 152 (1), 2007)

#### Synopsis

Essential Topology brings the most exciting b and useful - aspects of modern topology within reach of the average second-year undergraduate student. It contains all the essentials. The first chapter provides a complete account of continuity beginning at a level that a high school student could understand. The algebraic notions are introduced slowly through the text, leading the reader to the celebrated Hairy Ball theorem, and on to homotopy and homology b the cornerstones of contemporary algebraic topology.

Each topic is introduced with a thorough explanation of why it is being studied, and the focus throughout is on providing interesting examples that will motivate the student. Emphasis is placed on the basic objects that occur in research topology, and in its applications to other areas of mathematics.

This book is designed to provide a one-stop shop for undergraduate topology, providing enough material for two semester-long courses, and leaving students motivated and prepared for postgraduate study.

#### Synopsis

This book brings the most important aspects of modern topology within reach of a second-year undergraduate student. It successfully unites the most exciting aspects of modern topology with those that are most useful for research. The book is ideal for self-study.

#### Synopsis

This book brings the most important aspects of modern topology within reach of a second-year undergraduate student. It successfully unites the most exciting aspects of modern topology with those that are most useful for research, leaving readers prepared and motivated for further study. Written from a thoroughly modern perspective, every topic is introduced with an explanation of why it is being studied, and a huge number of examples provide further motivation. The book is ideal for self-study and assumes only a familiarity with the notion of continuity and basic algebra.

### Table of Contents

Introduction.- Continuous Functions.- Topological Spaces.- Interlude.- Topological Properties.- Deconstructionist Topology.- Interlude.- Homotopy.- The Euler Number.- Homotopy Groups.- Simplicial Homology.- Singular Homology.- More Deconstructionism.- Solutions to Selected Exercises.- Bibliography.- Index.