Set Spaces

This textbook presents the theory of Metric Spaces necessary for studying analysis beyond one real variable. Rich in examples, exercises and motivation, it provides a careful and clear exposition at a pace appropriate to the material.

This book examines an abstract mathematical theory, placing special emphasis on results applicable to formal logic. If a theory is especially abstract, it may find a natural home within several of the more familiar branches of mathematics. This is the case with the theory of closure spaces. It might be considered part of topology, lattice theory, universal algebra or, no doubt, one of several other branches of mathematics as well. In our development we have treated it, conceptually and methodologically, as part of topology, partly because we first thought ofthe basic structure involved (closure space), as a generalization of Frechet's concept V-space. V-spaces have been used in some developments of general topology as a generalization of topological space. Indeed, when in the early '50s, one of us started thinking about closure spaces, we thought ofit as the generalization of Frechet V­ space which comes from not requiring the null set to be CLOSURE SPACES ANDLOGIC XlI closed(as it is in V-spaces).

In this edition, a set of Supplementary Notes and Remarks has been added at the end, grouped according to chapter. Some of these call attention to subsequent developments, others add further explanation or additional remarks. Most of the remarks are accompanied by a briefly indicated proof, which is sometimes different from the one given in the reference cited.

The Minimal Piano Collection for one piano (BC8551) ended with “In C” by Terry Riley. Many of the composers on this new 11 Cd box have stated that this particular piece was a starting point for them to compose in a new way, exploring the ways that were created, the spaces between the notes, the emptiness, the freedom. Elizabeth Bergmann & Marcel Bergmann, Tamara Rumiantsev and Sandra van Veen & Jeroen van Veen recorded 83 compositions. From two pianos up to six pianos, from Steve Reich ‘standard’ Pianophase to Six Pianos. On the included Cd-ROM you can find all the biographies, background information and some sheet music. Van Veen recorded many pieces on his own using multi track technique.

In this book we study sprays and Finsler metrics. Roughly speaking, a spray on a manifold consists of compatible systems of second-order ordinary differential equations. A Finsler metric on a manifold is a family of norms in tangent spaces, which vary smoothly with the base point. Every Finsler metric determines a spray by its systems of geodesic equations. Thus, Finsler spaces can be viewed as special spray spaces. On the other hand, every Finsler metric defines a distance function by the length of minimial curves. Thus Finsler spaces can be viewed as regular metric spaces. Riemannian spaces are special regular metric spaces. In 1854, B. Riemann introduced the Riemann curvature for Riemannian spaces in his ground-breaking Habilitationsvortrag.