The Classical Fields. Structural Features of the Real and Rational Numbers

This book studies sets of rational and real numbers from the algebraic, ordered and topological points of view, both separately and with interrelations. The core of the book is in chapter 1, covering all the basic and deep results related to the reals. It treats R as an additive or multiplicative group, as an ordered set, a topological space (with its topological characterizations, groups of homeomorphisms, continuous images of R or of the Cantor set, and p-adic metrics), a field, an ordered group, a topological group, a measure space, an ordered field, and finally as a topological field. The next chapter defines non-standard rationals and reals using ultraproducts. Then comes a study of the rationals done in a similar way to the study of reals in chapter 1, and completions of ordered groups and fields, of topological groups, and of topological rings and fields. One chapter is devoted to p-adic numbers, again from the point of view of groups and fields, both with and without topologies. The appendix contains the basics of ordinal and cardinal numbers, of topological groups and of Pontryagin duality.

Most of the sections have exercises (with hints or solutions at the end of the book). The book ends with an extensive bibliography (about 350 items) and not too large an index. The book is certainly useful for readers with an interest in the interrelations of algebra and topology not based on abstract objects but on concrete ones, here reals and rationals.

Reviewer: 
mihus
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