To the reader
NEW EDITION
1. The Elements of Mathematics series takes up mathematics at the beginning and gives complete proofs. So in principle, it requires no particular knowledge of mathematics on the readers’ part, only a certain familiarity with mathematical reasoning and a certain capacity for abstract thinking. Nevertheless, it is directed primarily at those with a good knowledge of at least the content of the first year or two of university mathematics studies.
2. The chosen method of exposition is axiomatic and proceeds typically from the general to the specific. The requirements of proof impose a rigorously fixed order on the subject matter. It follows that the usefulness of certain considerations may not become apparent to readers until later chapters, unless they already have a fairly extended knowledge of mathematics.
3. The series is divided into Books and each Book into chapters. The Books that have already been published, either entirely or in part, in the French edition, are listed below. When an English translation is available, the corresponding English title is mentioned between parentheses. Throughout the volume, a reference indicates the English edition when available and the French edition otherwise.
Théorie des ensembles (Theory of Sets) Ref. E (Set Theory)
Algèbre (Algebra[^1]) — A (Alg.)
Topologie générale (General Topology) — TG (Gen. Top.)
Fonctions d’une variable réelle
(Functions in One Real Variable) — FVR (FRV)
[^1]: So far, only Chapters I through VIII have been translated. v
Espaces vectoriels topologiques
(Topological Vector Spaces) — EVT (Top. Vect. Sp.)
Intégration (Integration) — INT (Int.)
Algèbre commutative
(Commutative Algebra[^2]) — AC (Comm. Alg.)
Variétés différentiables et analytiques — VAR
Groupes et algèbres de Lie
(Lie Groups and Lie Algebras) — LIE (Lie)
Théories spectrales — TS
Topologie Algébrique — TA
In the first six Books (in the order mentioned above), every statement in the text assumes known only the definitions and results already discussed in the same chapters or in the previous chapters in the following order: Set Theory; Alg., Chapters I through III; Gen. Top., Chapters I through III; Alg./A, from Chapter IV on; Gen. Top., from Chapter IV on; FRV; Top. Vect. Sp.; Int. From the seventh Book on, the reader will find, if necessary, at the beginning of each Book or chapter, a precise indication of the other Books or chapters used (the first six Books are always assumed known).
4. However, certain passages do not follow the rules set out above. They are placed between two asterisks: . . . . In some cases, this is done only to
help readers understand the text by including examples that refer to results they may know from other sources. In other cases, we use not only results assumed known in the chapters in question but also results proved elsewhere in the series. These passages are used freely in the parts that assume known the chapters containing them and the chapters they refer to. We hope that the reader will be able to verify the absence of any vicious circle.
5. Some Books (either published or in preparation) may include summaries of results. These summaries contain the essential definitions and results of the Book, without any proofs.
6. The logical framework of each chapter consists of the chapter’s definitions, axioms, and theorems. These are the main parts to keep in mind for subsequent use. Less important results, and those that can be easily recovered from the theorems, are labeled as “propositions,” “lemmas,” “corollaries,” “remarks,” etc. Those that may be skipped at first reading are printed in a
[^2]: So far, only Chapters I through VII have been translated.
small font. A commentary on a fundamental theorem occasionally appears under the name of “scholium.”
To avoid tedious repetitions, it is sometimes convenient to introduce notation or abbreviations that are in force only within a specific chapter or section or subsection of a chapter (for example, in a chapter where all rings are commutative, the word “ring” may always refer to a “commutative ring”). Such conventions are mentioned explicitly at the beginning of the chapter, section, or subsection in which they apply.
Take note that, as in previous chapters of the present Book, fields are not assumed to be commutative unless stated otherwise.
7. Some passages are designed to forewarn the reader against serious errors. These passages are indicated in the margin with the sign (“dangerous bend”).
8. The exercises are meant, on the one hand, to allow readers to verify that they have digested the text well and, on the other hand, to let them acquaint themselves with results that have no place in the text. The most difficult exercises are marked with a .
9. We have paid particular attention to the terminology used in this series. We have adhered to the commonly accepted terminology except where there appeared to be good reasons to deviate from it.
10. We have made a particular effort to always use rigorously correct language, without sacrificing simplicity. As far as possible, we have drawn attention in the text to any abuses of language or notation, without which any mathematical text runs the risk of becoming pedant, not to say unreadable.
11. Since the text consists of the dogmatic exposition of a theory, it rarely contains any references to the literature. Bibliographical references are sometimes gathered together in Historical Notes. The bibliography that follows these Notes contains, in general, only those books and original memoirs that were of the greatest importance in the evolution of the theory under discussion. It does not pretend to be complete.
As to the exercises, we have not deemed it useful to indicate their origin since they have been taken from many different sources (original papers, textbooks, collections of exercises).
12. In this chapter of this Book of the new edition, references to theorems, axioms, definitions, remarks, etc. are given by indicating, successively, the Book (using the abbreviation listed in No. 3), chapter, section, subsection, and page where they can be found. The reference to the Book is left out when it is the same as the present Book. For example, in the Book Algebra,
Set Theory, III, § 4, No. 2, p. 167, Corollary 3
refers to Corollary 3 of § 4, No. 2 on page 167 of Chapter III of the Book Set Theory;
II, § 1, No. 11, p. 215, Proposition 17
refers to Proposition 17 of § 1, No. 11 on page 215 of Chapter II of the Book Algebra.
The summaries of results are indicated by the letter R; for example, Top. Vect. Sp., R refers to the “summary of results” of the Book Topological Vector Spaces.
As some Books will be published later in the new edition, references to those Books consist of, successively, the Book, chapter, section, and subsection where the results in question should be, for example,
Comm. Alg., III, § 4, No. 5, Corollary of Proposition 6.
When a reference to a volume of the French edition is made, the acronyms in upright capitals are used, and the French terminology and typography is used; for example,
TA, II, § 2, n, p. 158, corollaire de la proposition 1
refers to the corollary of Proposition 1 of § 2, No. 4 on page 158 of Chapter II of the Book Topologie Algébrique.