convergence foundations of topology

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Convergence Foundations Of Topology

Author : Szymon Dolecki
ISBN : 9789814571548
Genre : Mathematics
File Size : 39. 9 MB
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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 : C. Wayne Patty
ISBN : 9781449668655
Genre : Mathematics
File Size : 86. 69 MB
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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.

Mathematics Of Fuzzy Sets

Author : Ulrich Höhle
ISBN : 9781461550792
Genre : Mathematics
File Size : 88. 30 MB
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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.

Beyond Topology

Author : FrŽdŽric Mynard
ISBN : 9780821842799
Genre : Mathematics
File Size : 35. 34 MB
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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.

Elementary Topology

Author : Michael C. Gemignani
ISBN : 0486665224
Genre : Mathematics
File Size : 51. 85 MB
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Topology is one of the most rapidly expanding areas of mathematical thought: while its roots are in geometry and analysis, topology now serves as a powerful tool in almost every sphere of mathematical study. This book is intended as a first text in topology, accessible to readers with at least three semesters of a calculus and analytic geometry sequence. In addition to superb coverage of the fundamentals of metric spaces, topologies, convergence, compactness, connectedness, homotopy theory, and other essentials, Elementary Topology gives added perspective as the author demonstrates how abstract topological notions developed from classical mathematics. For this second edition, numerous exercises have been added as well as a section dealing with paracompactness and complete regularity. The Appendix on infinite products has been extended to include the general Tychonoff theorem; a proof of the Tychonoff theorem which does not depend on the theory of convergence has also been added in Chapter 7.

Foundations Of Free Noncommutative Function Theory

Author : Dmitry S. Kaliuzhnyi-Verbovetskyi
ISBN : 9781470416973
Genre : Mathematics
File Size : 58. 61 MB
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In this book the authors develop a theory of free noncommutative functions, in both algebraic and analytic settings. Such functions are defined as mappings from square matrices of all sizes over a module (in particular, a vector space) to square matrices over another module, which respect the size, direct sums, and similarities of matrices. Examples include, but are not limited to, noncommutative polynomials, power series, and rational expressions. Motivation and inspiration for using the theory of free noncommutative functions often comes from free probability. An important application area is "dimensionless" matrix inequalities; these arise, e.g., in various optimization problems of system engineering. Among other related areas are those of polynomial identities in rings, formal languages and finite automata, quasideterminants, noncommutative symmetric functions, operator spaces and operator algebras, and quantum control.

Implementing Cisco Ip Routing Route Foundation Learning Guide

Author : Diane Teare
ISBN : 9780133965865
Genre : Computers
File Size : 37. 63 MB
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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 [email protected] 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

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