This book is the second volume of a full and detailed course in the elements of real and complex analysis that mathematical undergraduates may expect to meet. Indeed, it was initially based on those parts of analysis that undergraduates at Cambridge University meet, or used to meet, in their first two years. There is however always a temptation to go a bit further, and this is a temptation that I have not resisted. Thus I have included accounts of Baire's category theorem, and the Arzela-Ascoli theorem, which are taught in the third year, and the Mazur–Ulam theorem, which, as far as I know, has never been taught. As a consequence, there are certain sections that can be omitted on a first reading. These are indicated by asterisks.

Volume I was concerned with analysis on the real line. In Part Three, the analysis is extended to a more general setting. We introduce and consider metric and topological spaces, and normed spaces. In fact, metric and metrizable spaces are sufficient for all subsequent needs, but many of the properties that we investigate are topological properties, and it is well worth understanding what this means. The study of topological spaces can degenerate into the construction of pathological examples; once again, temptation is not resisted, and Section 11.6 contains a collection of these. This section can be omitted at a first reading (and indeed at any subsequent reading).