sets, orderings, abstract algebra, MacLane-Eilenberg category theory, formal logic, general topology, uniform spaces, Baire category theory, Banach spaces and topological vector spaces, scalar and vector measures, Lebesgue and Henstock-Stieltjes integrals, fixed points, and differential equations in abstract spaces.Though HAF's main goal is to orchestrate classical material, the book also contains some results that cannot be easily found in other introductory books -- Bessaga's and Meyers's converses of the Contraction Fixed Point Theorem, Aarnes and Andenaes's redefinition of subnet, Gherman's characterization of topological convergences, Neumann's nonlinear Closed Graph Theorem, van Maaren's geometry-free version of Sperner's Lemma, etc.
The printed version of HAF is 883 + xxii pages long, with over 350 references; the CD-ROM version will be a little longer. Some of the excerpts below require a graphical browser such as Netscape.
A central theme of the book is the study of existence proofs. (An earlier, prepublication title of the book was CHOICE, COMPLETENESS, COMPACTNESS, for the three main ingredients of existence proofs.) Like most modern mathematics books, HAF is largely nonconstructive, and it includes the usual nonconstructive proofs of existence of pathological objects. Unlike most analysis books, however, HAF also includes a few pages discussing constructivism, choice, and consistency, to show that many of those classical pathological objects are inherently unconstructible. (For instance, every analyst knows that the set is nonempty, but HAF explains why we cannot construct an explicit example of a member of that set. The proof is based on a consistency result proved by Shelah in 1984.) I feel that when we cannot give an explicit example, we should say so; the student who cannot visualize some pathological object will be reassured to hear that no one else can visualize it either. Apparently I chose my words well; Jet Wimp (in S.I.A.M. Review) called my exposition "the most satisfying reflection on constructivism I have ever seen." Along with these discussions, HAF includes: