Convergence Foundations Of Topology PDF Download
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Author: Szymon Dolecki
Publisher: World Scientific Publishing Company
ISBN: 9814571547
Category : Mathematics
Languages : en
Pages : 569
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Book Description
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: World Scientific Publishing Company
ISBN: 9814571547
Category : Mathematics
Languages : en
Pages : 569
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Book Description
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: Gerhard Preuß
Publisher: Springer Science & Business Media
ISBN: 9401004897
Category : Mathematics
Languages : en
Pages : 306
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Book Description
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 Publishing Company Incorporated
ISBN: 9789814571517
Category : Mathematics
Languages : en
Pages : 548
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Book Description
It is clear that the book, with its high level of scholarship, is a labour of love on the part of its authors, and it will no doubt make quite a dent in the right circles ... The mature reader, disposed to this way of redoing topology, will have a great time." Mathematical Association of America 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:
ISBN: 9789814571531
Category :
Languages : en
Pages :
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Book Description
Author: Szymon Dolecki
Publisher: World Scientific
ISBN: 9811232121
Category : Mathematics
Languages : en
Pages : 733
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Book Description
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: Frdric Mynard
Publisher: American Mathematical Soc.
ISBN: 082184279X
Category : Mathematics
Languages : en
Pages : 395
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Book Description
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: William J. Pervin
Publisher: Academic Press
ISBN: 1483225151
Category : Mathematics
Languages : en
Pages : 222
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Book Description
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: Tai-Danae Bradley
Publisher: MIT Press
ISBN: 0262359626
Category : Mathematics
Languages : en
Pages : 167
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Book Description
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: Ulrich Höhle
Publisher: Springer Science & Business Media
ISBN: 146151617X
Category : Mathematics
Languages : en
Pages : 377
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Book Description
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: Tim Maudlin
Publisher:
ISBN: 0198701306
Category : Mathematics
Languages : en
Pages : 374
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Book Description
Tim Maudlin sets out a completely new method for describing the geometrical structure of spaces, and thus a better mathematical tool for describing and understanding space-time. He presents a historical review of the development of geometry and topology, and then his original Theory of Linear Structures.
