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.
General
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