Preface -- 1 Introduction -- 2 The Real Line -- 3 Topology of Euclidean Spaces -- 4 Measure Theory -- 5 Useful Tools and Technologies -- 6 Descriptive Set Theory -- 7 Decline and Fall of the Duality -- 8 Special Sets of Reals -- 9 Additional Axioms -- 10 Undecidable Statements -- 11 Appendix -- Bibliography -- Index of Notation -- Index.

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.