I first came across Sutherland’s Topological Spaces sometime in 2003 – about a year before I started my Maths degree. Back then I didn’t know very much about Mathematics, let alone any Mathematics itself, and I certainly didn’t have the maturity to grasp most of what was being discussed. Five years later, I’m thankful that I decided to continue reading …
Overview: The book is a standard first course in point-set topology. It can roughly be divided into two parts. The first half of the book is motivated by the desire to generalize the familiar notion of continuity for real valued functions. The reader is first introduced to metric spaces and then to the more general notion of a toplogical space.
The second half discusses various special properties of topological spaces such as compactness and connectedness and the interactions between them. Towards the end Sutherland returns to metric spaces to study convergence of sequences in a more general setting.
The Good: While the author dosen’t shy away from hard mathematical formalism, he usually provides plenty of motivation and discussion beforehand to ease the reader into the more tricky ideas. The careful reader will rarely feel lost. Everything in the book fits together extremely tightly, and the pace never gets overwhelming even for the more casual Mathematics student. I also liked the choice of examples, which illustrate the theory but aren’t overly convoluted.
The exercises are pitched at the right level for a beginner – Most of them are not very hard yet steer clear of being completely trivial. The inclusion of a Hints section makes the book ideal for self study.
The Bad: My main (and only) complaint is regarding the numbering system. References like Proposition 3.7.28 are simply not very helpful.
In Closing: Topological Spaces is a text that every undergraduate should read at least once. Despite the rather annoying numbering system, the book completely immerses the reader in the subject. Indeed those who persevere through the text will not only gain a first class introduction to point-set topology but also a greater appreciation of mathematics as a whole.
Introduction to Metric and Topological Spaces is published by Oxford University Press. (ISBN: 0-19-853161-3)
