\input zb-basic
\input zb-matheduc
\iteman{ZMATH 2014e.00633}
\itemau{Stillwell, John}
\itemti{The real numbers. An introduction to set theory and analysis.}
\itemso{Undergraduate Texts in Mathematics. Cham: Springer (ISBN 978-3-319-01576-7/hbk; 978-3-319-01577-4/ebook). xvi, 244~p. (2013).}
\itemab
Despite its subtitle this book will be fully appreciated by either professional mathematicians or those students, who already have passed a course in analysis or set theory. It is not the case that the text cannot be used as the first reading. It can, but the added value here provides a wider overview above the real numbers theory and links to other subjects -- mostly mathematical, but also historical. The introductory chapter is an excellent example of the art how to make the text attractive -- its subsections are e.g. Why does $ab=ba$? What are numbers? What is a line? What is geometry? What are functions? What is continuity? What is a measure? These questions are in different aspects discussed later in the book and summarized again in the last chapter. The study of the reals and their properties is done in close connection with the study of related concepts from analysis, as it is obvious from the chapter titles: From discrete to continuous, Infinite sets, Functions and limits, Open sets and continuity, Ordinals, The axiom of choice, Borel sets, Measure theory. Therefore calculus is an absolutely necessary prerequisite for the reader. The book contains a quantity of motivation examples, worked-out examples and exercises, what makes it suitable also for self-study.
\itemrv{Vladim\'\i r Jani\v s (Bansk\'a Bystrica)}
\itemcc{I15 E65}
\itemut{real number; set theory; axiom of choice; ordinals; continuity; measurability}
\itemli{doi:10.1007/978-3-319-01577-4}
\end