The Structure of the Real Line

1. Front Matter

   Pages i-xiv

   PDF

2. Introduction

   • Lev Bukovský

   Pages 1-38

3. The Real Line

   • Lev Bukovský

   Pages 39-71

4. Metric Spaces and Real Functions

   • Lev Bukovský

   Pages 73-125

5. Measure Theory

   • Lev Bukovský

   Pages 127-159

6. Useful Tools and Technologies

   • Lev Bukovský

   Pages 161-214

7. Descriptive Set Theory

   • Lev Bukovský

   Pages 215-248

8. Decline and Fall of the Duality

   • Lev Bukovský

   Pages 249-291

9. Special Sets of Reals

   • Lev Bukovský

   Pages 293-371

10. Additional Axioms

    • Lev Bukovský

    Pages 373-414

11. Undecidable Statements

    • Lev Bukovský

    Pages 415-454

12. Appendix

    • Lev Bukovský

    Pages 455-491

13. Back Matter

    Pages 493-536

    PDF

The rapid development of set theory in the last fifty years, mainly by obtaining plenty of independence results, strongly influenced an understanding of the structure of the real line. This book is devoted to the study of the real line and its subsets taking into account the recent results of set theory. Whenever possible the presentation is done without the full axiom of choice. Since the book is intended to be self-contained, all necessary results of set theory, topology, measure theory, and descriptive set theory are revisited with the purpose of eliminating superfluous use of an axiom of choice. The duality of measure and category is studied in a detailed manner. Several statements pertaining to properties of the real line are shown to be undecidable in set theory. The metamathematics behind set theory is shortly explained in the appendix. Each section contains a series of exercises with additional results.

• measure theory
• set theory
• set topology

From the reviews:

“The history of investigations of real numbers is the starting point of this monograph. … Every chapter is supplied with a well-chosen series of exercises containing interesting complementary material. … This unique monograph presents beautiful scientific work prepared in a professional style and built upon a number of references to prominent mathematicians living in previous epochs … . A wide variety of researchers and students will find in it a large … attractive body of knowledge that should be highly inspiring in their work.” (Marek Balcerzak, Mathematical Reviews, Issue 2012 d)

“The book under review is addressed to experts and begins to address some of the questions about the complexities of the real line. … The author provides an appendix with background material on set theory, algebra, topology and metamathematics.” (William J. Satzer, The Mathematical Association of America, August, 2011)

• , Institute of Mathematics, University of P.J. Šafárik, Košice, Slovakia

  Lev Bukovský

Policies and ethics