Basic concepts of mathematics. Set-theoretic, algebraic, topological foundations as well as real and complex numbers (Q2361500)

In the 1990s and thereafter, the four-volume treatise ``Textbook of mathematics. For mathematicians, computer scientists and physicists'' by the German authors U. Storch (1940--2017) and H. Wiebe enjoyed a great popularity at German-language universities. This series incorporated volumes on real analysis of one variable, linear algebra, analysis of several real variables and integration theory as well as on the topics of analysis on manifolds, complex function theory and functional analysis, respectively. Now, almost thirty years later, the authors have begun a new edition of this well-established treatise, together with their German colleagues C. Becker (Wiesbaden), M. Kersken (Flensburg) and F. Loose (Tübingen). Also, the entire material of the former textbook series will be given an essentially new structure, on the one hand, and a substantial enlargement of the treated topics on the other. Basically, the planned thirteen volumes will treat more areas in greater depth and more specifically, and the single volumes will therefore be less voluminous than the four books of the former series. More precisely, the following books of the new series are provided for the future: 1. Basic concepts of mathematics; 2. Analysis of one variable; 3. Linear algebra I; 4. Linear algebra II; 5. Differential calculus of several variables; 6. Measure and integration theory; 7. Analysis on manifolds; 8. Functional analysis; 9. Stochastics; 10. Ordinary differential equations; 11. Complex function theory; 12. Differential geometry and differential topology; 13. Algebra. The book under review is the announced first volume of the anticipated new series, written by the founding authors U. Storch and H. Wiebe, and just finished before U. Storch's untimely decease in 2017. This book is to serve as a general basis for the subsequent twelve volumes, in that it covers some fundamental concepts of set theory, abstract algebra, general topology, and number systems. In particular, this preparatory part fixes the overall terminology, provides numerous basic concepts and results from the afore-mentioned branches of mathematics, and systematically collects various miscellanies scattered over the earlier textbook series. Chapter 1 treats the basics of set theory, including the natural number system, finite sets and combinatorics, infinite sets and cardinal numbers, ordinal numbers, and the prime factorization of natural numbers. Chapter~2 is devoted to those algebraic fundamentals such as monoids and groups, monoid and group actions on sets, permutation groups, rings, ideals and factor rings, modules and vector spaces, algebras, and principal ideal domains as well as factorial rings. Chapter~3 treats real and complex numbers, where the basic approach is via general ordered fields, convergent sequences therein, real number fields, and the concept of completeness. After the description of the field of complex numbers, the general properties of series and their summability, the basics of continuous functions (on compact sets), and those real functions such as exponential, logarithms, and power functions are discussed. Topological structures are the main topic of the concluding Chapter~4, where metric spaces, normed vector spaces, general topological spaces and continuous maps, connectedness and compactness of topological spaces, and the concept of uniform convergence in complete metric spaces are explained in great detail. The entire text is accompanied by numerous instructive examples, inspiring outlooks, further-leading remarks, and carefully selected exercises, partly with hints. Altogether, this book perfectly provides what its title promises, namely a solid introduction to some of the most fundamental concepts and key results in modern university mathematics. The universal approach presented here may serve as a very useful, general source for the study of different textbooks of any kind or purpose. In the meantime, the second volume in this new series has already appeared under the title ``Analysis of one variable. Analytic function, differentiation and integration'' (German) [\textit{U. Storch} and \textit{H. Wiebe}, Analysis einer Veränderlichen. Analytische Funktionen, Differenziation und Integration. Berlin: Springer Spektrum (2018; Zbl 1403.26003)], and here it becomes evident how much this part is based on the first volume under review, and how efficient the authors' general new approach is.