Elements of Mathematics

This book is a reprint of the 1966 edition. It completes the treatment of general topology begun in chapters I - IV (also available in an English translation). The real numbers having been introduced in chapter IV, the first chapters of this volume study subgroups and quotients of R (with applications to the "measurement of magnitudes" and to the log and exp functions), then real vector spaces and projective spaces, then the additive groups Rn (subgroups, quotients, homomorphisms, infinite sums and products). Analogous properties are then studied for complex numbers, in chapters VIII. Chapter IX illustrates the use of real numbers in general topology, studying different important kinds of topological spaces: uniformizable, metric, normal Baire, Polish, Borel. The final chapter deals with the various topologies of function spaces, ending with a section on approximation of functions. The book also includes a historical note and numerous exercises.