© 2011

The Structure of the Real Line


Part of the Monografie Matematyczne book series (MONOGRAFIE, volume 71)

Table of contents

  1. Front Matter
    Pages i-xiv
  2. Lev Bukovský
    Pages 1-38
  3. Lev Bukovský
    Pages 39-71
  4. Lev Bukovský
    Pages 73-125
  5. Lev Bukovský
    Pages 127-159
  6. Lev Bukovský
    Pages 161-214
  7. Lev Bukovský
    Pages 215-248
  8. Lev Bukovský
    Pages 249-291
  9. Lev Bukovský
    Pages 293-371
  10. Lev Bukovský
    Pages 373-414
  11. Lev Bukovský
    Pages 415-454
  12. Lev Bukovský
    Pages 455-491
  13. Back Matter
    Pages 493-536

About this book


The rapid development of set theory in the last fifty years, mainly in 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, descriptive set theory are revisited with the purpose to eliminate 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 it is shortly explained in the appendix. Each section contains a series of exercises with additional results.


measure theory set theory set topology

Authors and affiliations

  1. 1., Institute of MathematicsUniversity of P.J. ŠafárikKošiceSlovakia

Bibliographic information

  • Book Title The Structure of the Real Line
  • Authors Lev Bukovský
  • Series Title Monografie Matematyczne
  • DOI
  • Copyright Information Springer Basel AG 2011
  • Publisher Name Springer, Basel
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Hardcover ISBN 978-3-0348-0005-1
  • Softcover ISBN 978-3-0348-0321-2
  • eBook ISBN 978-3-0348-0006-8
  • Edition Number 1
  • Number of Pages XIV, 542
  • Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
  • Topics Analysis
  • Buy this book on publisher's site
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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)