Convergence Foundations of Topology

Author: Szymon Dolecki
Publisher: World Scientific Publishing Company
ISBN: 9814571547
Format: PDF, Mobi
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This textbook is an alternative to a classical introductory book in point-set topology. The approach, however, is radically different from the classical one. It is based on convergence rather than on open and closed sets. Convergence of filters is a natural generalization of the basic and well-known concept of convergence of sequences, so that convergence theory is more natural and intuitive to many, perhaps most, students than classical topology. On the other hand, the framework of convergence is easier, more powerful and far-reaching which highlights a need for a theory of convergence in various branches of analysis. Convergence theory for filters is gradually introduced and systematically developed. Topological spaces are presented as a special subclass of convergence spaces of particular interest, but a large part of the material usually developed in a topology textbook is treated in the larger realm of convergence spaces.

Foundations of Topology

Author: Gerhard Preuß
Publisher: Springer Science & Business Media
ISBN: 9401004897
Format: PDF
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A new foundation of Topology, summarized under the name Convenient Topology, is considered such that several deficiencies of topological and uniform spaces are remedied. This does not mean that these spaces are superfluous. It means exactly that a better framework for handling problems of a topological nature is used. In this setting semiuniform convergence spaces play an essential role. They include not only convergence structures such as topological structures and limit space structures, but also uniform convergence structures such as uniform structures and uniform limit space structures, and they are suitable for studying continuity, Cauchy continuity and uniform continuity as well as convergence structures in function spaces, e.g. simple convergence, continuous convergence and uniform convergence. Various interesting results are presented which cannot be obtained by using topological or uniform spaces in the usual context. The text is self-contained with the exception of the last chapter, where the intuitive concept of nearness is incorporated in Convenient Topology (there exist already excellent expositions on nearness spaces).

Foundations of General Topology

Author: William J. Pervin
Publisher: Academic Press
ISBN: 1483225151
Format: PDF, Mobi
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Foundations of General Topology presents the value of careful presentations of proofs and shows the power of abstraction. This book provides a careful treatment of general topology. Organized into 11 chapters, this book begins with an overview of the important notions about cardinal and ordinal numbers. This text then presents the fundamentals of general topology in logical order processing from the most general case of a topological space to the restrictive case of a complete metric space. Other chapters consider a general method for completing a metric space that is applicable to the rationals and present the sufficient conditions for metrizability. This book discusses as well the study of spaces of real-valued continuous functions. The final chapter deals with uniform continuity of functions, which involves finding a distance that satisfies certain requirements for all points of the space simultaneously. This book is a valuable resource for students and research workers.

Foundations of Topology

Author: C. Wayne Patty
Publisher: Jones & Bartlett Publishers
ISBN: 1449668658
Format: PDF, Kindle
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Topology is a branch of pure mathematics that deals with the abstract relationships found in geometry and analysis. Written with the mature student in mind, Foundations of Topology, Second Edition, provides a user-friendly, clear, and concise introduction to this fascinating area of mathematics. The author introduces topics that are well motivated with thorough proofs that make them easy to follow. Historical comments are dispersed throughout the text, and exercises, varying in degree of difficulty, are found at the end of each chapter. Foundations of Topology is an excellent text for teaching students how to develop the skill to write clear and precise proofs.

The Foundations of Topological Analysis A Straightforward Introduction

Author: K. G. Binmore
Publisher: CUP Archive
ISBN: 9780521299305
Format: PDF, Kindle
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This book is an introduction to the ideas from general topology that are used in elementary analysis. It is written at a level that is intended to make the bulk of the material accessible to students in the latter part of their first year of study at a university or college although students will normally meet most of the work in their second or later years. The aim has been to bridge the gap between introductory books like the author's Mathematical Analysis: A Straightforward Approach, in which carefully selected theorems are discussed at length with numerous examples, and the more advanced book on analysis, in which the author is more concerned with providing a comprehensive and elegant theory than in smoothing the ways for beginners. An attempt has been made throughout not only to prepare the ground for more advanced work, but also to revise and to illuminate the material which students will have met previously but may have not fully understood.

Encyclopedia of General Topology

Author: K.P. Hart
Publisher: Elsevier
ISBN: 9780080530864
Format: PDF, ePub, Docs
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This book is designed for the reader who wants to get a general view of the terminology of General Topology with minimal time and effort. The reader, whom we assume to have only a rudimentary knowledge of set theory, algebra and analysis, will be able to find what they want if they will properly use the index. However, this book contains very few proofs and the reader who wants to study more systematically will find sufficiently many references in the book. Key features: • More terms from General Topology than any other book ever published • Short and informative articles • Authors include the majority of top researchers in the field • Extensive indexing of terms

Handbook of Analysis and Its Foundations

Author: Eric Schechter
Publisher: Academic Press
ISBN: 9780080532998
Format: PDF, ePub, Docs
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Handbook of Analysis and Its Foundations is a self-contained and unified handbook on mathematical analysis and its foundations. Intended as a self-study guide for advanced undergraduates and beginning graduatestudents in mathematics and a reference for more advanced mathematicians, this highly readable book provides broader coverage than competing texts in the area. Handbook of Analysis and Its Foundations provides an introduction to a wide range of topics, including: algebra; topology; normed spaces; integration theory; topological vector spaces; and differential equations. The author effectively demonstrates the relationships between these topics and includes a few chapters on set theory and logic to explain the lack of examples for classical pathological objects whose existence proofs are not constructive. More complete than any other book on the subject, students will find this to be an invaluable handbook. Covers some hard-to-find results including: Bessagas and Meyers converses of the Contraction Fixed Point Theorem Redefinition of subnets by Aarnes and Andenaes Ghermans characterization of topological convergences Neumanns nonlinear Closed Graph Theorem van Maarens geometry-free version of Sperners Lemma Includes a few advanced topics in functional analysis Features all areas of the foundations of analysis except geometry Combines material usually found in many different sources, making this unified treatment more convenient for the user Has its own webpage: http://math.vanderbilt.edu/

General Topology

Author: John L. Kelley
Publisher: Courier Dover Publications
ISBN: 0486820661
Format: PDF, ePub, Mobi
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Comprehensive text for beginning graduate-level students and professionals. "The clarity of the author's thought and the carefulness of his exposition make reading this book a pleasure." — Bulletin of the American Mathematical Society. 1955 edition.

Foundations of Topology

Author: CTI Reviews
Publisher: Cram101 Textbook Reviews
ISBN: 161906555X
Format: PDF, Kindle
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Facts101 is your complete guide to Foundations of Topology. In this book, you will learn topics such as as those in your book plus much more. With key features such as key terms, people and places, Facts101 gives you all the information you need to prepare for your next exam. Our practice tests are specific to the textbook and we have designed tools to make the most of your limited study time.

Mathematics of Fuzzy Sets

Author: Ulrich Höhle
Publisher: Springer Science & Business Media
ISBN: 1461550793
Format: PDF
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Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory is a major attempt to provide much-needed coherence for the mathematics of fuzzy sets. Much of this book is new material required to standardize this mathematics, making this volume a reference tool with broad appeal as well as a platform for future research. Fourteen chapters are organized into three parts: mathematical logic and foundations (Chapters 1-2), general topology (Chapters 3-10), and measure and probability theory (Chapters 11-14). Chapter 1 deals with non-classical logics and their syntactic and semantic foundations. Chapter 2 details the lattice-theoretic foundations of image and preimage powerset operators. Chapters 3 and 4 lay down the axiomatic and categorical foundations of general topology using lattice-valued mappings as a fundamental tool. Chapter 3 focuses on the fixed-basis case, including a convergence theory demonstrating the utility of the underlying axioms. Chapter 4 focuses on the more general variable-basis case, providing a categorical unification of locales, fixed-basis topological spaces, and variable-basis compactifications. Chapter 5 relates lattice-valued topologies to probabilistic topological spaces and fuzzy neighborhood spaces. Chapter 6 investigates the important role of separation axioms in lattice-valued topology from the perspective of space embedding and mapping extension problems, while Chapter 7 examines separation axioms from the perspective of Stone-Cech-compactification and Stone-representation theorems. Chapters 8 and 9 introduce the most important concepts and properties of uniformities, including the covering and entourage approaches and the basic theory of precompact or complete [0,1]-valued uniform spaces. Chapter 10 sets out the algebraic, topological, and uniform structures of the fundamentally important fuzzy real line and fuzzy unit interval. Chapter 11 lays the foundations of generalized measure theory and representation by Markov kernels. Chapter 12 develops the important theory of conditioning operators with applications to measure-free conditioning. Chapter 13 presents elements of pseudo-analysis with applications to the Hamilton–Jacobi equation and optimization problems. Chapter 14 surveys briefly the fundamentals of fuzzy random variables which are [0,1]-valued interpretations of random sets.