# Search Results for "topology-of-metric-spaces"

## Topology of Metric Spaces

**Author**: S. Kumaresan**Publisher:**Alpha Science Int'l Ltd.**ISBN:**9781842652503**Category:**Mathematics**Page:**152**View:**459

"Topology of Metric Spaces gives a very streamlined development of a course in metric space topology emphasizing only the most useful concepts, concrete spaces and geometric ideas to encourage geometric thinking, to treat this as a preparatory ground for a general topology course, to use this course as a surrogate for real analysis and to help the students gain some perspective of modern analysis." "Eminently suitable for self-study, this book may also be used as a supplementary text for courses in general (or point-set) topology so that students will acquire a lot of concrete examples of spaces and maps."--BOOK JACKET.

## Metric Spaces

**Author**: Satish Shirali,Harkrishan Lal Vasudeva**Publisher:**Springer Science & Business Media**ISBN:**9781852339227**Category:**Mathematics**Page:**222**View:**5586

Since the last century, the postulational method and an abstract point of view have played a vital role in the development of modern mathematics. The experience gained from the earlier concrete studies of analysis point to the importance of passage to the limit. The basis of this operation is the notion of distance between any two points of the line or the complex plane. The algebraic properties of underlying sets often play no role in the development of analysis; this situation naturally leads to the study of metric spaces. The abstraction not only simplifies and elucidates mathematical ideas that recur in different guises, but also helps eco- mize the intellectual effort involved in learning them. However, such an abstract approach is likely to overlook the special features of particular mathematical developments, especially those not taken into account while forming the larger picture. Hence, the study of particular mathematical developments is hard to overemphasize. The language in which a large body of ideas and results of functional analysis are expressed is that of metric spaces. The books on functional analysis seem to go over the preliminaries of this topic far too quickly. The present authors attempt to provide a leisurely approach to the theory of metric spaces. In order to ensure that the ideas take root gradually but firmly, a large number of examples and counterexamples follow each definition. Also included are several worked examples and exercises. Applications of the theory are spread out over the entire book.

## Introduction to General Topology

**Author**: K. D. Joshi**Publisher:**New Age International**ISBN:**9780852264447**Category:**Topology**Page:**412**View:**1852

## Metric Spaces

**Author**: Mícheál O'Searcoid**Publisher:**Springer Science & Business Media**ISBN:**9781846286278**Category:**Mathematics**Page:**304**View:**5832

The abstract concepts of metric spaces are often perceived as difficult. This book offers a unique approach to the subject which gives readers the advantage of a new perspective on ideas familiar from the analysis of a real line. Rather than passing quickly from the definition of a metric to the more abstract concepts of convergence and continuity, the author takes the concrete notion of distance as far as possible, illustrating the text with examples and naturally arising questions. Attention to detail at this stage is designed to prepare the reader to understand the more abstract ideas with relative ease.

## Introduction to Metric and Topological Spaces

**Author**: Wilson Alexander Sutherland,W. A. Sutherland**Publisher:**Oxford University Press**ISBN:**9780198531616**Category:**Science**Page:**181**View:**7137

One of the ways in which topology has influenced other branches of mathematics in the past few decades is by putting the study of continuity and convergence into a general setting. This book introduces metric and topological spaces by describing some of that influence. The aim is to move gradually from familiar real analysis to abstract topological spaces. The book is aimed primarily at the second-year mathematics student, and numerous exercises are included.

## Set Theory and Metric Spaces

**Author**: Irving Kaplansky**Publisher:**American Mathematical Soc.**ISBN:**0821826948**Category:**Mathematics**Page:**140**View:**4548

This is a book that could profitably be read by many graduate students or by seniors in strong major programs ... has a number of good features. There are many informal comments scattered between the formal development of theorems and these are done in a light and pleasant style. ... There is a complete proof of the equivalence of the axiom of choice, Zorn's Lemma, and well-ordering, as well as a discussion of the use of these concepts. There is also an interesting discussion of the continuum problem ... The presentation of metric spaces before topological spaces ... should be welcomed by most students, since metric spaces are much closer to the ideas of Euclidean spaces with which they are already familiar. --Canadian Mathematical Bulletin Kaplansky has a well-deserved reputation for his expository talents. The selection of topics is excellent. -- Lance Small, UC San Diego This book is based on notes from a course on set theory and metric spaces taught by Edwin Spanier, and also incorporates with his permission numerous exercises from those notes. The volume includes an Appendix that helps bridge the gap between metric and topological spaces, a Selected Bibliography, and an Index.

## Elementary Topology

**Author**: Michael C. Gemignani**Publisher:**Courier Corporation**ISBN:**9780486665221**Category:**Mathematics**Page:**270**View:**9407

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.

## Elements of Metric Spaces

**Author**: N.A**Publisher:**Academic Publishers**ISBN:**9788189781989**Category:**Metric spaces**Page:**204**View:**824

## Topology

*Point-Set and Geometric*

**Author**: Paul L. Shick**Publisher:**John Wiley & Sons**ISBN:**9781118030585**Category:**Mathematics**Page:**296**View:**873

The essentials of point-set topology, complete with motivation andnumerous examples Topology: Point-Set and Geometric presents an introduction totopology that begins with the axiomatic definition of a topology ona set, rather than starting with metric spaces or the topology ofsubsets of Rn. This approach includes many more examples, allowingstudents to develop more sophisticated intuition and enabling themto learn how to write precise proofs in a brand-new context, whichis an invaluable experience for math majors. Along with the standard point-set topologytopics—connected and path-connected spaces, compact spaces,separation axioms, and metric spaces—Topology covers theconstruction of spaces from other spaces, including products andquotient spaces. This innovative text culminates with topics fromgeometric and algebraic topology (the Classification Theorem forSurfaces and the fundamental group), which provide instructors withthe opportunity to choose which "capstone" best suits his or herstudents. Topology: Point-Set and Geometric features: A short introduction in each chapter designed to motivate theideas and place them into an appropriate context Sections with exercise sets ranging in difficulty from easy tofairly challenging Exercises that are very creative in their approaches and workwell in a classroom setting A supplemental Web site that contains complete and colorfulillustrations of certain objects, several learning modulesillustrating complicated topics, and animations of particularlycomplex proofs

## Introduction to the Analysis of Metric Spaces

**Author**: John R. Giles,John Robilliard Giles**Publisher:**Cambridge University Press**ISBN:**9780521359283**Category:**Mathematics**Page:**257**View:**6730

Assuming a basic knowledge of real analysis and linear algebra, the student is given some familiarity with the axiomatic method in analysis and is shown the power of this method in exploiting the fundamental analysis structures underlying a variety of applications. Although the text is titled metric spaces, normed linear spaces are introduced immediately because this added structure is present in many examples and its recognition brings an interesting link with linear algebra; finite dimensional spaces are discussed earlier. It is intended that metric spaces be studied in some detail before general topology is begun. This follows the teaching principle of proceeding from the concrete to the more abstract. Graded exercises are provided at the end of each section and in each set the earlier exercises are designed to assist in the detection of the abstract structural properties in concrete examples while the latter are more conceptually sophisticated.