Convergence Foundations Of Topology

Author : Szymon Dolecki,Frédéric Mynard
Publisher : World Scientific Publishing Company
Page : 568 pages
File Size : 54,8 Mb
Release : 2016-05-13
Category : Mathematics
ISBN : 9789814571548

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Convergence Foundations of Topology by Szymon Dolecki,Frédéric Mynard Pdf

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.

Author : Szymon Dolecki
Publisher : Unknown
Page : 128 pages
File Size : 43,7 Mb
Release : 2016
Category : Electronic
ISBN : 9814571539

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Convergence Foundations of Topology by Szymon Dolecki Pdf

Author : Gerhard Preuß
Publisher : Springer Science & Business Media
Page : 303 pages
File Size : 54,5 Mb
Release : 2011-06-27
Category : Mathematics
ISBN : 9789401004893

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Foundations of Topology by Gerhard Preuß Pdf

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).

Author : Szymon Dolecki
Publisher : World Scientific
Page : 733 pages
File Size : 41,8 Mb
Release : 2024-01-03
Category : Mathematics
ISBN : 9789811232121

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Royal Road To Topology, A: Convergence Of Filters by Szymon Dolecki Pdf

Topological spaces are a special case of convergence spaces. This textbook introduces topology within a broader context of convergence theory. The title alludes to advantages of the present approach, which is more gratifying than many traditional ones: you travel more comfortably through mathematical landscapes and you see more.The book is addressed both to those who wish to learn topology and to those who, being already knowledgeable about topology, are curious to review it from a different perspective, which goes well beyond the traditional knowledge.Usual topics of classic courses of set-theoretic topology are treated at an early stage of the book — from a viewpoint of convergence of filters, but in a rather elementary way. Later on, most of these facts reappear as simple consequences of more advanced aspects of convergence theory.The mentioned virtues of the approach stem from the fact that the class of convergences is closed under several natural, essential operations, under which the class of topologies is not! Accordingly, convergence theory complements topology like the field of complex numbers algebraically completes the field of real numbers.Convergence theory is intuitive and operational because of appropriate level of its abstraction, general enough to grasp the underlying laws, but not too much in order not to lose intuitive appeal.

Author : C. Wayne Patty
Publisher : Jones & Bartlett Learning
Page : 398 pages
File Size : 54,5 Mb
Release : 2009
Category : Computers
ISBN : 9780763742348

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Foundations of Topology by C. Wayne Patty Pdf

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.

Author : John Wilder Tukey
Publisher : Unknown
Page : 128 pages
File Size : 40,8 Mb
Release : 1965
Category : Electronic
ISBN : OCLC:878090248

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Convergence and uniformity in topology by John Wilder Tukey Pdf

Author : Tai-Danae Bradley,Tyler Bryson,John Terilla
Publisher : MIT Press
Page : 167 pages
File Size : 50,6 Mb
Release : 2020-08-18
Category : Mathematics
ISBN : 9780262359627

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Topology by Tai-Danae Bradley,Tyler Bryson,John Terilla Pdf

A graduate-level textbook that presents basic topology from the perspective of category theory. This graduate-level textbook on topology takes a unique approach: it reintroduces basic, point-set topology from a more modern, categorical perspective. Many graduate students are familiar with the ideas of point-set topology and they are ready to learn something new about them. Teaching the subject using category theory--a contemporary branch of mathematics that provides a way to represent abstract concepts--both deepens students' understanding of elementary topology and lays a solid foundation for future work in advanced topics.

Author : William J. Pervin
Publisher : Academic Press
Page : 222 pages
File Size : 54,6 Mb
Release : 2014-05-12
Category : Mathematics
ISBN : 9781483225159

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Foundations of General Topology by William J. Pervin Pdf

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.

Author : John W. Tukey
Publisher : Unknown
Page : 89 pages
File Size : 50,5 Mb
Release : 1958
Category : Electronic
ISBN : OCLC:66380924

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Convergence and Uniformity in Topology by John W. Tukey Pdf

Author : FrŽdŽric Mynard,Elliott Pearl
Publisher : American Mathematical Soc.
Page : 395 pages
File Size : 45,8 Mb
Release : 2009-05-15
Category : Mathematics
ISBN : 9780821842799

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Beyond Topology by FrŽdŽric Mynard,Elliott Pearl Pdf

The purpose of this collection is to guide the non-specialist through the basic theory of various generalizations of topology, starting with clear motivations for their introduction. Structures considered include closure spaces, convergence spaces, proximity spaces, quasi-uniform spaces, merotopic spaces, nearness and filter spaces, semi-uniform convergence spaces, and approach spaces. Each chapter is self-contained and accessible to the graduate student, and focuses on motivations to introduce the generalization of topologies considered, presenting examples where desirable properties are not present in the realm of topologies and the problem is remedied in the more general context. Then, enough material will be covered to prepare the reader for more advanced papers on the topic. While category theory is not the focus of the book, it is a convenient language to study these structures and, while kept as a tool rather than an object of study, will be used throughout the book. For this reason, the book contains an introductory chapter on categorical topology.

Author : John L. Kelley
Publisher : Courier Dover Publications
Page : 321 pages
File Size : 41,7 Mb
Release : 2017-03-17
Category : Mathematics
ISBN : 9780486815442

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General Topology by John L. Kelley Pdf

"The clarity of the author's thought and the carefulness of his exposition make reading this book a pleasure," noted the Bulletin of the American Mathematical Society upon the 1955 publication of John L. Kelley's General Topology. This comprehensive treatment for beginning graduate-level students immediately found a significant audience, and it remains a highly worthwhile and relevant book for students of topology and for professionals in many areas. A systematic exposition of the part of general topology that has proven useful in several branches of mathematics, this volume is especially intended as background for modern analysis. An extensive preliminary chapter presents mathematical foundations for the main text. Subsequent chapters explore topological spaces, the Moore-Smith convergence, product and quotient spaces, embedding and metrization, and compact, uniform, and function spaces. Each chapter concludes with an abundance of problems, which form integral parts of the discussion as well as reinforcements and counter examples that mark the boundaries of possible theorems. The book concludes with an extensive index that provides supplementary material on elementary set theory.

Author : Ulrich Höhle
Publisher : Springer Science & Business Media
Page : 402 pages
File Size : 54,5 Mb
Release : 2001-04-30
Category : Mathematics
ISBN : 0792373189

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Many Valued Topology and its Applications by Ulrich Höhle Pdf

The 20th Century brought the rise of General Topology. It arose from the effort to establish a solid base for Analysis and it is intimately related to the success of set theory. Many Valued Topology and Its Applications seeks to extend the field by taking the monadic axioms of general topology seriously and continuing the theory of topological spaces as topological space objects within an almost completely ordered monad in a given base category C. The richness of this theory is shown by the fundamental fact that the category of topological space objects in a complete and cocomplete (epi, extremal mono)-category C is topological over C in the sense of J. Adamek, H. Herrlich, and G.E. Strecker. Moreover, a careful, categorical study of the most important topological notions and concepts is given - e.g., density, closedness of extremal subobjects, Hausdorff's separation axiom, regularity, and compactness. An interpretation of these structures, not only by the ordinary filter monad, but also by many valued filter monads, underlines the richness of the explained theory and gives rise to new concrete concepts of topological spaces - so-called many valued topological spaces. Hence, many valued topological spaces play a significant role in various fields of mathematics - e.g., in the theory of locales, convergence spaces, stochastic processes, and smooth Borel probability measures. In its first part, the book develops the necessary categorical basis for general topology. In the second part, the previously given categorical concepts are applied to monadic settings determined by many valued filter monads. The third part comprises various applications of many valued topologies to probability theory and statistics as well as to non-classical model theory. These applications illustrate the significance of many valued topology for further research work in these important fields.

Author : Jean-Charles Pinoli
Publisher : John Wiley & Sons
Page : 397 pages
File Size : 48,9 Mb
Release : 2014-07-22
Category : Technology & Engineering
ISBN : 9781118984550

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Mathematical Foundations of Image Processing and Analysis, Volume 2 by Jean-Charles Pinoli Pdf

Mathematical Imaging is currently a rapidly growing field in applied mathematics, with an increasing need for theoretical mathematics. This book, the second of two volumes, emphasizes the role of mathematics as a rigorous basis for imaging sciences. It provides a comprehensive and convenient overview of the key mathematical concepts, notions, tools and frameworks involved in the various fields of gray-tone and binary image processing and analysis, by proposing a large, but coherent, set of symbols and notations, a complete list of subjects and a detailed bibliography. It establishes a bridge between the pure and applied mathematical disciplines, and the processing and analysis of gray-tone and binary images. It is accessible to readers who have neither extensive mathematical training, nor peer knowledge in Image Processing and Analysis. It is a self-contained book focusing on the mathematical notions, concepts, operations, structures, and frameworks that are beyond or involved in Image Processing and Analysis. The notations are simplified as far as possible in order to be more explicative and consistent throughout the book and the mathematical aspects are systematically discussed in the image processing and analysis context, through practical examples or concrete illustrations. Conversely, the discussed applicative issues allow the role of mathematics to be highlighted. Written for a broad audience – students, mathematicians, image processing and analysis specialists, as well as other scientists and practitioners – the author hopes that readers will find their own way of using the book, thus providing a mathematical companion that can help mathematicians become more familiar with image processing and analysis, and likewise, image processing and image analysis scientists, researchers and engineers gain a deeper understanding of mathematical notions and concepts.

Author : Diane Teare,Bob Vachon,Rick Graziani
Publisher : Cisco Press
Page : 765 pages
File Size : 45,6 Mb
Release : 2014-12-29
Category : Computers
ISBN : 9780133965865

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Implementing Cisco IP Routing (ROUTE) Foundation Learning Guide by Diane Teare,Bob Vachon,Rick Graziani Pdf

Now updated for Cisco’s new ROUTE 300-101 exam, Implementing Cisco IP Routing (ROUTE) Foundation Learning Guide is your Cisco® authorized learning tool for CCNP® or CCDP® preparation. Part of the Cisco Press Foundation Learning Series, it teaches you how to plan, configure, maintain, and scale a modern routed network. Focusing on Cisco routers connected in LANs and WANs at medium-to-large network sites, the authors show how to select and implement Cisco IOS services for building scalable, routed networks. They examine basic network and routing protocol principles in detail; introduce both IPv4 and IPv6; fully review EIGRP, OSPF, and BGP; explore enterprise Internet connectivity; cover routing updates and path control; and present today’s router security best practices. Each chapter opens with a list of topics that clearly identifies its focus. Each chapter ends with a summary of key concepts for quick study, as well as review questions to assess and reinforce your understanding. Throughout, configuration and verification output examples illustrate critical issues in network operation and troubleshooting. This guide is ideal for all certification candidates who want to master all the topics covered on the ROUTE 300-101 exam. Serves as the official book for the newest version of the Cisco Networking Academy CCNP ROUTE course Includes all the content from the newest Learning@Cisco ROUTE course and information on each of the ROUTE exam topics Compares basic routing protocol features and limitations Examines RIPv2 and RIPng Covers EIGRP operation and implementation for both IPv4 and IPv6 Explores OSPFv2 implementation, and OSPFv3 for both IPv4 and IPv6 Discusses network performance optimization via routing updates Introduces path control with Cisco Express Forwarding (CEF) switching, policy-based routing (PBR), and service level agreements (SLAs) Addresses enterprise Internet connectivity via single or redundant ISP connections Explains BGP terminology, concepts, operation, configuration, verification, and troubleshooting Covers securing the management plane of Cisco routers using authentication and other recommended practices Presents self-assessment review questions, chapter objectives, and summaries to facilitate effective studying

Author : Eric Schechter
Publisher : Academic Press
Page : 907 pages
File Size : 43,7 Mb
Release : 1996-10-24
Category : Mathematics
ISBN : 9780080532998

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Handbook of Analysis and Its Foundations by Eric Schechter Pdf

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/