Hausdorff on Ordered Sets

Hausdorff on Ordered Sets
Title Hausdorff on Ordered Sets PDF eBook
Author Felix Hausdorff
Publisher American Mathematical Soc.
Pages 343
Release 2005
Genre Mathematics
ISBN 0821837885

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Georg Cantor, the founder of set theory, published his last paper on sets in 1897. In 1900, David Hilbert made Cantor's Continuum Problem and the challenge of well-ordering the real numbers the first problem in his famous Paris lecture. It was time for the appearance of the second generation of Cantorians. They emerged in the decade 1900-1909, and foremost among them were Ernst Zermelo and Felix Hausdorff. Zermelo isolated the Choice Principle, proved that every set could be well-ordered, and axiomatized the concept of set. He became the father of abstract set theory. Hausdorff eschewed foundations and pursued set theory as part of the mathematical arsenal. He was recognized as the era's leading Cantorian. From 1901-1909, Hausdorff published seven articles in which he created a representation theory for ordered sets and investigated sets of real sequences partially ordered by eventual dominance, together with their maximally ordered subsets. These papers are translated and appear in this volume. Each is accompanied by an introductory essay. These highly accessible works are of historical significance, not only for set theory, but also for model theory, analysis and algebra.

Set Theory

Set Theory
Title Set Theory PDF eBook
Author Felix Hausdorff
Publisher American Mathematical Soc.
Pages 352
Release 2021-08-24
Genre Education
ISBN 1470464942

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This work is a translation into English of the Third Edition of the classic German language work Mengenlehre by Felix Hausdorff published in 1937. From the Preface (1937): “The present book has as its purpose an exposition of the most important theorems of the theory of sets, along with complete proofs, so that the reader should not find it necessary to go outside this book for supplementary details while, on the other hand, the book should enable him to undertake a more detailed study of the voluminous literature on the subject. The book does not presuppose any mathematical knowledge beyond the differential and integral calculus, but it does require a certain maturity in abstract reasoning; qualified college seniors and first year graduate students should have no difficulty in making the material their own … The mathematician will … find in this book some things that will be new to him, at least as regards formal presentation and, in particular, as regards the strengthening of theorems, the simplification of proofs, and the removal of unnecessary hypotheses.”

Non-Hausdorff Topology and Domain Theory

Non-Hausdorff Topology and Domain Theory
Title Non-Hausdorff Topology and Domain Theory PDF eBook
Author Jean Goubault-Larrecq
Publisher Cambridge University Press
Pages 499
Release 2013-03-28
Genre Mathematics
ISBN 1107328772

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This unique book on modern topology looks well beyond traditional treatises and explores spaces that may, but need not, be Hausdorff. This is essential for domain theory, the cornerstone of semantics of computer languages, where the Scott topology is almost never Hausdorff. For the first time in a single volume, this book covers basic material on metric and topological spaces, advanced material on complete partial orders, Stone duality, stable compactness, quasi-metric spaces and much more. An early chapter on metric spaces serves as an invitation to the topic (continuity, limits, compactness, completeness) and forms a complete introductory course by itself. Graduate students and researchers alike will enjoy exploring this treasure trove of results. Full proofs are given, as well as motivating ideas, clear explanations, illuminating examples, application exercises and some more challenging problems for more advanced readers.

Sets and Extensions in the Twentieth Century

Sets and Extensions in the Twentieth Century
Title Sets and Extensions in the Twentieth Century PDF eBook
Author
Publisher Elsevier
Pages 878
Release 2012-01-24
Genre Mathematics
ISBN 0080930662

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Set theory is an autonomous and sophisticated field of mathematics that is extremely successful at analyzing mathematical propositions and gauging their consistency strength. It is as a field of mathematics that both proceeds with its own internal questions and is capable of contextualizing over a broad range, which makes set theory an intriguing and highly distinctive subject. This handbook covers the rich history of scientific turning points in set theory, providing fresh insights and points of view. Written by leading researchers in the field, both this volume and the Handbook as a whole are definitive reference tools for senior undergraduates, graduate students and researchers in mathematics, the history of philosophy, and any discipline such as computer science, cognitive psychology, and artificial intelligence, for whom the historical background of his or her work is a salient consideration - Serves as a singular contribution to the intellectual history of the 20th century - Contains the latest scholarly discoveries and interpretative insights

An Introduction to Partially Ordered Structures and Sheaves

An Introduction to Partially Ordered Structures and Sheaves
Title An Introduction to Partially Ordered Structures and Sheaves PDF eBook
Author Francisco Miraglia
Publisher Polimetrica s.a.s.
Pages 517
Release 2006
Genre Mathematics
ISBN 8876990356

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Counterexamples in Topology

Counterexamples in Topology
Title Counterexamples in Topology PDF eBook
Author Lynn Arthur Steen
Publisher Courier Corporation
Pages 274
Release 2013-04-22
Genre Mathematics
ISBN 0486319296

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Over 140 examples, preceded by a succinct exposition of general topology and basic terminology. Each example treated as a whole. Numerous problems and exercises correlated with examples. 1978 edition. Bibliography.

Topologies on Closed and Closed Convex Sets

Topologies on Closed and Closed Convex Sets
Title Topologies on Closed and Closed Convex Sets PDF eBook
Author Gerald Beer
Publisher Springer Science & Business Media
Pages 360
Release 1993-10-31
Genre Mathematics
ISBN 9780792325314

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This monograph provides an introduction to the theory of topologies defined on the closed subsets of a metric space, and on the closed convex subsets of a normed linear space as well. A unifying theme is the relationship between topology and set convergence on the one hand, and set functionals on the other. The text includes for the first time anywhere an exposition of three topologies that over the past ten years have become fundamental tools in optimization, one-sided analysis, convex analysis, and the theory of multifunctions: the Wijsman topology, the Attouch--Wets topology, and the slice topology. Particular attention is given to topologies on lower semicontinuous functions, especially lower semicontinuous convex functions, as associated with their epigraphs. The interplay between convex duality and topology is carefully considered and a chapter on set-valued functions is included. The book contains over 350 exercises and is suitable as a graduate text. This book is of interest to those working in general topology, set-valued analysis, geometric functional analysis, optimization, convex analysis and mathematical economics.