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Books in General topology

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Topological Vector Spaces, Distributions and Kernels

  • 1st Edition
  • February 16, 2016
  • François Treves
  • Paul A. Smith + 1 more
  • English
  • eBook
    9 7 8 - 1 - 4 8 3 2 - 2 3 6 2 - 9
Topological Vector Spaces, Distributions and Kernels discusses partial differential equations involving spaces of functions and space distributions. The book reviews the definitions of a vector space, of a topological space, and of the completion of a topological vector space. The text gives examples of Frechet spaces, Normable spaces, Banach spaces, or Hilbert spaces. The theory of Hilbert space is similar to finite dimensional Euclidean spaces in which they are complete and carry an inner product that can determine their properties. The text also explains the Hahn-Banach theorem, as well as the applications of the Banach-Steinhaus theorem and the Hilbert spaces. The book discusses topologies compatible with a duality, the theorem of Mackey, and reflexivity. The text describes nuclear spaces, the Kernels theorem and the nuclear operators in Hilbert spaces. Kernels and topological tensor products theory can be applied to linear partial differential equations where kernels, in this connection, as inverses (or as approximations of inverses), of differential operators. The book is suitable for vector mathematicians, for students in advanced mathematics and physics.

Handbook of Set-Theoretic Topology

  • 1st Edition
  • June 28, 2014
  • K. Kunen + 1 more
  • English
  • eBook
    9 7 8 - 1 - 4 8 3 2 - 9 5 1 5 - 2
This Handbook is an introduction to set-theoretic topology for students in the field and for researchers in other areas for whom results in set-theoretic topology may be relevant. The aim of the editors has been to make it as self-contained as possible without repeating material which can easily be found in standard texts. The Handbook contains detailed proofs of core results, and references to the literature for peripheral results where space was insufficient. Included are many open problems of current interest.In general, the articles may be read in any order. In a few cases they occur in pairs, with the first one giving an elementary treatment of a subject and the second one more advanced results. These pairs are: Hodel and Juhász on cardinal functions; Roitman and Abraham-Todorčević on S- and L-spaces; Weiss and Baumgartner on versions of Martin's axiom; and Vaughan and Stephenson on compactness properties.

Modern General Topology

  • 2nd Edition
  • May 12, 2014
  • Jun-Iti Nagata
  • N. G. De Bruijn + 2 more
  • English
  • eBook
    9 7 8 - 1 - 4 8 3 2 - 7 8 1 6 - 2
Bibliotheca Mathematica: A Series of Monographs on Pure and Applied Mathematics, Volume VII: Modern General Topology focuses on the processes, operations, principles, and approaches employed in pure and applied mathematics, including spaces, cardinal and ordinal numbers, and mappings. The publication first elaborates on set, cardinal and ordinal numbers, basic concepts in topological spaces, and various topological spaces. Discussions focus on metric space, axioms of countability, compact space and paracompact space, normal space and fully normal space, subspace, product space, quotient space, and inverse limit space, convergence, mapping, and open basis and neighborhood basis. The book then ponders on compact spaces and related topics, as well as product of compact spaces, compactification, extensions of the concept of compactness, and compact space and the lattice of continuous functions. The manuscript tackles paracompact spaces and related topics, metrizable spaces and related topics, and topics related to mappings. Topics include metric space, paracompact space, and continuous mapping, theory of inverse limit space, theory of selection, mapping space, imbedding, metrizability, uniform space, countably paracompact space, and modifications of the concept of paracompactness. The book is a valuable source of data for mathematicians and researchers interested in modern general topology.

Foundations of General Topology

  • 1st Edition
  • May 12, 2014
  • William J. Pervin
  • Ralph P. Boas
  • English
  • eBook
    9 7 8 - 1 - 4 8 3 2 - 2 5 1 5 - 9
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.

Generalized Functions

  • 1st Edition
  • May 12, 2014
  • I. M. Gel'fand + 1 more
  • English
  • eBook
    9 7 8 - 1 - 4 8 3 2 - 6 2 2 4 - 6
Generalized Functions, Volume 4: Applications of Harmonic Analysis is devoted to two general topics—developments in the theory of linear topological spaces and construction of harmonic analysis in n-dimensional Euclidean and infinite-dimensional spaces. This volume specifically discusses the bilinear functionals on countably normed spaces, Hilbert-Schmidt operators, and spectral analysis of operators in rigged Hilbert spaces. The general form of positive generalized functions on the space S, continuous positive-definite functions, and conditionally positive generalized functions are also deliberated. This publication likewise considers the mean of a generalized random process, multidimensional generalized random fields, simplest properties of cylinder sets, and definition of Gaussian measures. This book is beneficial to students, specialists, and researchers aiming to acquire knowledge of functional analysis.

Geometric Topology

  • 1st Edition
  • May 10, 2014
  • James C. Cantrell
  • English
  • eBook
    9 7 8 - 1 - 4 8 3 2 - 7 1 3 1 - 6
Geometric Topology contains the proceedings of the 1977 Georgia Topology Conference, held at the University of Georgia on August 1977. The book is comprised of contributions from leading experts in the field of geometric topology.These contributions are grouped into four sections: low dimensional manifolds, topology of manifolds, shape theory and infinite dimensional topology, and miscellaneous problems. Subjects discussed under these sections include local spanning missing loops, the structure of generalized manifolds having nonmanifold set of trivial dimension, universal open principal fibrations, and how to build a flexible polyhedral surface. Topologists, geometers, and mathematicians will find the book very interesting and insightful.

Algebraical and Topological Foundations of Geometry

  • 1st Edition
  • May 9, 2014
  • Hans Freudenthal
  • English
  • eBook
    9 7 8 - 1 - 4 8 3 1 - 8 4 6 4 - 7
Algebraical and Topological Foundations of Geometry contains the proceedings of the Colloquium on Algebraic and Topological Foundations of Geometry, held in Utrecht, the Netherlands in August 1959. The papers review the algebraical and topological foundations of geometry and cover topics ranging from the geometric algebra of the Möbius plane to the theory of parallels with applications to closed geodesies. Groups of homeomorphisms and topological descriptive planes are also discussed. Comprised of 26 chapters, this book introduces the reader to the theory of parallels with applications to closed geodesies; groups of homeomorphisms; complemented modular lattices; and topological descriptive planes. Subsequent chapters focus on collineation groups; exceptional algebras and exceptional groups; the connection between algebra and constructions with ruler and compasses; and the use of differential geometry and analytic group theory methods in foundations of geometry. Von Staudt projectivities of Moufang planes are also considered, and an axiomatic treatment of polar geometry is presented. This monograph will be of interest to students of mathematics.

Explorations in Topology

  • 2nd Edition
  • December 4, 2013
  • David Gay
  • English
  • eBook
    9 7 8 - 0 - 1 2 - 4 1 6 6 4 0 - 0
Explorations in Topology, Second Edition, provides students a rich experience with low-dimensional topology (map coloring, surfaces, and knots), enhances their geometrical and topological intuition, empowers them with new approaches to solving problems, and provides them with experiences that will help them make sense of future, more formal topology courses. The book's innovative story-line style models the problem-solving process, presents the development of concepts in a natural way, and engages students in meaningful encounters with the material. The updated end-of-chapter investigations provide opportunities to work on many open-ended, non-routine problems and, through a modified "Moore method," to make conjectures from which theorems emerge. The revised end-of-chapter notes provide historical background to the chapter's ideas, introduce standard terminology, and make connections with mainstream mathematics. The final chapter of projects provides ideas for continued research. Explorations in Topology, Second Edition, enhances upper division courses and is a valuable reference for all levels of students and researchers working in topology.

Saks Spaces and Applications to Functional Analysis

  • 2nd Edition
  • Volume 139
  • August 18, 2011
  • J.B. Cooper
  • English
  • eBook
    9 7 8 - 0 - 0 8 - 0 8 7 2 5 0 - 6
The first edition of this monograph appeared in 1978. In view of the progress made in the intervening years, the original text has been revised, several new sections have been added and the list of references has been updated. The book presents a systematic treatment of the theory of Saks Spaces, i.e. vector space with a norm and related, subsidiary locally convex topology. Applications are given to space of bounded, continuous functions, to measure theory, vector measures, spaces of bounded measurable functions, spaces of bounded analytic functions, and to W*-algebras.

Open Problems in Topology II

  • 1st Edition
  • March 15, 2007
  • Elliott M. Pearl
  • English
  • eBook
    9 7 8 - 0 - 0 8 - 0 4 7 5 2 9 - 5
This volume is a collection of surveys of research problems in topology and its applications. The topics covered include general topology, set-theoretic topology, continuum theory, topological algebra, dynamical systems, computational topology and functional analysis.