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Title Topological Properties of Spaces of Continuous Functions
Author Robert A. McCoy
Publisher Springer
Release 2006-12-08
Category Mathematics
Total Pages 130
ISBN 3540391819
Language English, Spanish, and French
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Book Summary:

This book brings together into a general setting various techniques in the study of the topological properties of spaces of continuous functions. The two major classes of function space topologies studied are the set-open topologies and the uniform topologies. Where appropriate, the analogous theorems for the two major classes of topologies are studied together, so that a comparison can be made. A chapter on cardinal functions puts characterizations of a number of topological properties of function spaces into a more general setting: some of these results are new, others are generalizations of known theorems. Excercises are included at the end of each chapter, covering other kinds of function space topologies. Thus the book should be appropriate for use in a classroom setting as well as for functional analysis and general topology. The only background needed is some basic knowledge of general topology.

Title Topological Properties of the Standard Operations on Spaces of Continuous Functions and Integrable Functions
Author Holly Michelle Renaud
Publisher
Release 2019
Category
Total Pages 0
ISBN
Language English, Spanish, and French
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Book Summary:

In this work, we consider different notions of openness for several operations on a large collection of different settings. We start with the scalar multiplication on sequence spaces, spaces of continuous functions, and integrable functions. We apply different techniques to derive weak-openness of multiplication on spaces of differentiable functions, endowed with a large collection of quasi-algebra norms. We consider the openness property of the dense-defined standard multiplication on spaces of integrable functions, as multiplication on these settings is only defined on a dense set. We adapt the definition of openness for the multiplication to include dense-defined products and then prove that the multiplication on these spaces, restricted to their corresponding domains, is uniformly open. We investigate sufficient and necessary conditions for pairs of functions to be points of local openness for the multiplication on normed spaces of differentiable functions. Next, we establish openness for other kinds of operations, more precisely, the maximum and minimum operations on spaces of scalar-valued integrable functions from a topological space into a finite measure on an algebra of subsets of that space. The last section of this work deals with the connection between openness of the multiplication on spaces of continuous functions and topological properties of the domain of those functions..

Title Topics in General Topology
Author K. Morita
Publisher Elsevier
Release 1989-08-04
Category Mathematics
Total Pages 746
ISBN 9780080879888
Language English, Spanish, and French
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Book Summary:

Being an advanced account of certain aspects of general topology, the primary purpose of this volume is to provide the reader with an overview of recent developments. The papers cover basic fields such as metrization and extension of maps, as well as newly-developed fields like categorical topology and topological dynamics. Each chapter may be read independently of the others, with a few exceptions. It is assumed that the reader has some knowledge of set theory, algebra, analysis and basic general topology.

Real Analysis by N. L. Carothers

Title Real Analysis
Author N. L. Carothers
Publisher Cambridge University Press
Release 2000-08-15
Category Mathematics
Total Pages 420
ISBN 9780521497565
Language English, Spanish, and French
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Book Summary:

A text for a first graduate course in real analysis for students in pure and applied mathematics, statistics, education, engineering, and economics.

Title The Infinite Dimensional Topology of Function Spaces
Author J. van Mill
Publisher Elsevier
Release 2002-05-24
Category Mathematics
Total Pages 642
ISBN 9780080929774
Language English, Spanish, and French
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Book Summary:

In this book we study function spaces of low Borel complexity. Techniques from general topology, infinite-dimensional topology, functional analysis and descriptive set theory are primarily used for the study of these spaces. The mix of methods from several disciplines makes the subject particularly interesting. Among other things, a complete and self-contained proof of the Dobrowolski-Marciszewski-Mogilski Theorem that all function spaces of low Borel complexity are topologically homeomorphic, is presented. In order to understand what is going on, a solid background in infinite-dimensional topology is needed. And for that a fair amount of knowledge of dimension theory as well as ANR theory is needed. The necessary material was partially covered in our previous book `Infinite-dimensional topology, prerequisites and introduction'. A selection of what was done there can be found here as well, but completely revised and at many places expanded with recent results. A `scenic' route has been chosen towards the Dobrowolski-Marciszewski-Mogilski Theorem, linking the results needed for its proof to interesting recent research developments in dimension theory and infinite-dimensional topology. The first five chapters of this book are intended as a text for graduate courses in topology. For a course in dimension theory, Chapters 2 and 3 and part of Chapter 1 should be covered. For a course in infinite-dimensional topology, Chapters 1, 4 and 5. In Chapter 6, which deals with function spaces, recent research results are discussed. It could also be used for a graduate course in topology but its flavor is more that of a research monograph than of a textbook; it is therefore more suitable as a text for a research seminar. The book consequently has the character of both textbook and a research monograph. In Chapters 1 through 5, unless stated otherwise, all spaces under discussion are separable and metrizable. In Chapter 6 results for more general classes of spaces are presented. In Appendix A for easy reference and some basic facts that are important in the book have been collected. The book is not intended as a basis for a course in topology; its purpose is to collect knowledge about general topology. The exercises in the book serve three purposes: 1) to test the reader's understanding of the material 2) to supply proofs of statements that are used in the text, but are not proven there 3) to provide additional information not covered by the text. Solutions to selected exercises have been included in Appendix B. These exercises are important or difficult.

General Topology III by A. V. Arhangel' skii

Title General Topology III
Author A. V. Arhangel' skii
Publisher Springer Science & Business Media
Release 2013-03-09
Category Mathematics
Total Pages 232
ISBN 3662074133
Language English, Spanish, and French
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Book Summary:

This reference work deals with important topics in general topology and their role in functional analysis and axiomatic set theory, for graduate students and researchers working in topology, functional analysis, set theory and probability theory. It provides a guide to recent research findings, with three contributions by Arhangel'skii and Choban.

Title Encyclopedia of General Topology
Author K.P. Hart
Publisher Elsevier
Release 2003-11-18
Category Mathematics
Total Pages 536
ISBN 9780080530864
Language English, Spanish, and French
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Book Summary:

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

Title Recent Progress in General Topology
Author M. Husek
Publisher Elsevier
Release 1992-11-20
Category Mathematics
Total Pages 808
ISBN 0080934439
Language English, Spanish, and French
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Book Summary:

These papers survey the developments in General Topology and the applications of it which have taken place since the mid 1980s. The book may be regarded as an update of some of the papers in the Handbook of Set-Theoretic Topology (eds. Kunen/Vaughan, North-Holland, 1984), which gives an almost complete picture of the state of the art of Set Theoretic Topology before 1984. In the present volume several important developments are surveyed that surfaced in the period 1984-1991. This volume may also be regarded as a partial update of Open Problems in Topology (eds. van Mill/Reed, North-Holland, 1990). Solutions to some of the original 1100 open problems are discussed and new problems are posed.

Title Introduction to General Topology
Author K. D. Joshi
Publisher New Age International
Release 1983
Category Topology
Total Pages 430
ISBN 9780852264447
Language English, Spanish, and French
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Book Summary:

Title Function Spaces with Uniform Fine and Graph Topologies
Author Robert A. McCoy
Publisher Springer
Release 2018-04-21
Category Mathematics
Total Pages 106
ISBN 3319770543
Language English, Spanish, and French
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Book Summary:

This book presents a comprehensive account of the theory of spaces of continuous functions under uniform, fine and graph topologies. Besides giving full details of known results, an attempt is made to give generalizations wherever possible, enriching the existing literature. The goal of this monograph is to provide an extensive study of the uniform, fine and graph topologies on the space C(X,Y) of all continuous functions from a Tychonoff space X to a metric space (Y,d); and the uniform and fine topologies on the space H(X) of all self-homeomorphisms on a metric space (X,d). The subject matter of this monograph is significant from the theoretical viewpoint, but also has applications in areas such as analysis, approximation theory and differential topology. Written in an accessible style, this book will be of interest to researchers as well as graduate students in this vibrant research area.