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This textbook in point set topology is aimed at an upper-undergraduate audience. Its gentle pace will be useful to students who are still learning to write proofs. Prerequisites include calculus and at least one semester of analysis, where the student has been properly exposed to the ideas of basic set theory such as subsets, unions, intersections, and functions, as well as convergence and other topological notions in the real line. Appendices are included to bridge the gap between this new material and material found in an analysis course. Metric spaces are one of the more prevalent topological spaces used in other areas and are therefore introduced in the first chapter and emphasized throughout the text. This also conforms to the approach of the book to start with the particular and work toward the more general. Chapter 2 defines and develops abstract topological spaces, with metric spaces as the source of inspiration, and with a focus on Hausdorff spaces. The final chapter concentrates on continuous real-valued functions, culminating in a development of paracompact spaces.
Book Synopsis A Course in Point Set Topology by : John B. Conway
Download or read book A Course in Point Set Topology written by John B. Conway and published by Springer Science & Business Media. This book was released on 2013-11-04 with total page 154 pages. Available in PDF, EPUB and Kindle. Book excerpt: This textbook in point set topology is aimed at an upper-undergraduate audience. Its gentle pace will be useful to students who are still learning to write proofs. Prerequisites include calculus and at least one semester of analysis, where the student has been properly exposed to the ideas of basic set theory such as subsets, unions, intersections, and functions, as well as convergence and other topological notions in the real line. Appendices are included to bridge the gap between this new material and material found in an analysis course. Metric spaces are one of the more prevalent topological spaces used in other areas and are therefore introduced in the first chapter and emphasized throughout the text. This also conforms to the approach of the book to start with the particular and work toward the more general. Chapter 2 defines and develops abstract topological spaces, with metric spaces as the source of inspiration, and with a focus on Hausdorff spaces. The final chapter concentrates on continuous real-valued functions, culminating in a development of paracompact spaces.
Suitable for a complete course in topology, this text also functions as a self-contained treatment for independent study. Additional enrichment materials make it equally valuable as a reference. 1964 edition.
Book Synopsis Point Set Topology by : Steven A. Gaal
Download or read book Point Set Topology written by Steven A. Gaal and published by Courier Corporation. This book was released on 2009-04-23 with total page 338 pages. Available in PDF, EPUB and Kindle. Book excerpt: Suitable for a complete course in topology, this text also functions as a self-contained treatment for independent study. Additional enrichment materials make it equally valuable as a reference. 1964 edition.
Topology continues to be a topic of prime importance in contemporary mathematics, but until the publication of this book there were few if any introductions to topology for undergraduates. This book remedied that need by offering a carefully thought-out, graduated approach to point set topology at the undergraduate level. To make the book as accessible as possible, the author approaches topology from a geometric and axiomatic standpoint; geometric, because most students come to the subject with a good deal of geometry behind them, enabling them to use their geometric intuition; axiomatic, because it parallels the student's experience with modern algebra, and keeps the book in harmony with current trends in mathematics. After a discussion of such preliminary topics as the algebra of sets, Euler-Venn diagrams and infinite sets, the author takes up basic definitions and theorems regarding topological spaces (Chapter 1). The second chapter deals with continuous functions (mappings) and homeomorphisms, followed by two chapters on special types of topological spaces (varieties of compactness and varieties of connectedness). Chapter 5 covers metric spaces. Since basic point set topology serves as a foundation not only for functional analysis but also for more advanced work in point set topology and algebraic topology, the author has included topics aimed at students with interests other than analysis. Moreover, Dr. Baum has supplied quite detailed proofs in the beginning to help students approaching this type of axiomatic mathematics for the first time. Similarly, in the first part of the book problems are elementary, but they become progressively more difficult toward the end of the book. References have been supplied to suggest further reading to the interested student.
Book Synopsis Elements of Point Set Topology by : John D. Baum
Download or read book Elements of Point Set Topology written by John D. Baum and published by Courier Corporation. This book was released on 1991-01-01 with total page 164 pages. Available in PDF, EPUB and Kindle. Book excerpt: Topology continues to be a topic of prime importance in contemporary mathematics, but until the publication of this book there were few if any introductions to topology for undergraduates. This book remedied that need by offering a carefully thought-out, graduated approach to point set topology at the undergraduate level. To make the book as accessible as possible, the author approaches topology from a geometric and axiomatic standpoint; geometric, because most students come to the subject with a good deal of geometry behind them, enabling them to use their geometric intuition; axiomatic, because it parallels the student's experience with modern algebra, and keeps the book in harmony with current trends in mathematics. After a discussion of such preliminary topics as the algebra of sets, Euler-Venn diagrams and infinite sets, the author takes up basic definitions and theorems regarding topological spaces (Chapter 1). The second chapter deals with continuous functions (mappings) and homeomorphisms, followed by two chapters on special types of topological spaces (varieties of compactness and varieties of connectedness). Chapter 5 covers metric spaces. Since basic point set topology serves as a foundation not only for functional analysis but also for more advanced work in point set topology and algebraic topology, the author has included topics aimed at students with interests other than analysis. Moreover, Dr. Baum has supplied quite detailed proofs in the beginning to help students approaching this type of axiomatic mathematics for the first time. Similarly, in the first part of the book problems are elementary, but they become progressively more difficult toward the end of the book. References have been supplied to suggest further reading to the interested student.
This book is an introduction to point set topology for undergraduates. Many of the classic textbooks on the subject cover the subject exhaustively and at the highest possible level of generality. The result of using traditional textbooks has been that students spend 2 semesters learning far more general topology on abstract spaces then most of them will ever need to use or know. More importantly, students get the impression from geometers and topologists in later courses that they "wasted" a year of their studies learning material that most mathematicians don't even consider topology anymore. This leaves many of them feeling deceived and frustrated. Unfortunately, the reaction has been in recent decades to write elementary topology textbooks that only present the barest minimum of point set topology needed for students in advanced geometry or algebraic topology. Indeed-some recent beginning textbooks in topology largely skip general topology altogether and jump straight into algebraic and geometric topology such as homotopy, curves and surfaces! We believe this ludicrous solution is essentially throwing the baby out with the bathwater. This reissued edition of Hall/ Spencer should seriously be considered by mathematicians as the benchmark for such a course. The book contains what we believe to be approximately the irreducible minimum of point set topology any student of mathematics needs to learn regardless of level or interest. The book is quite detailed, covering sufficient general topology of interest and use for analysts, geometers and topologists. The book falls into two rather distinct parts. The first half is concerned with an introductory study of topological and metric spaces. The basic operations with sets are introduced in Chapter I, relations and mappings are discussed, and an introduction to infinite and uncountable sets is given. Chapter 2 introduces the basic topological structure of the real numbers in a review of basic analysis. In Chapter 3, general topological and metric spaces are introduced and such topics as compactness, separation and continuous functions are discussed. Metric spaces are pursued further in Chapter 4, with discussions of local connectivity, countability, metrizability and completion being included. The second part is less elementary in character. The long Chapter 5 is concerned with giving topological characterizations of arcs, simple closed curves, and simple closed surfaces. Peano spaces are discussed and the Jordan curve theorem and Jordan-Schoenflies theorem are proved. Chapter 6 discusses partitionable spaces, a topic often missing from modern texts. Finally, Chapter 7 discusses the axiom of choice, Zorn's lemma (in the form commonly called the Hausdorff niaximality principle) and the Tychonoff product theorem. The book in particular will help students understand the deep connection between general topology and real and complex analysis. The most natural path towards understanding abstract topological spaces, general continuous mappings and topological invariants on families of open sets is to see how they directly generalize the usual structures of analysis on the real line. Also. Blue Collar Scholar founder/editor Karo Maestro has added his usual personal touch to the new edition, with a new preface on his own reflections on point set topology and recommendations for supplementary or subsequent study. The prerequisites for the text are very minimal-just calculus and some experience with rigorous proofs. This wonderful lost text in this new inexpensive edition will serve a new generation of mathematics students who need to learn this crucial foundational subject with a presentation that's both detailed and informative without being exhaustive. It will indoctrinate students into the beauty and simplicity of point-set topology and convince them of its' intrinsic importance-primarily to analysis, but also to other areas of mathematics.
Book Synopsis Elementary Topology by : Guilford Spencer
Download or read book Elementary Topology written by Guilford Spencer and published by . This book was released on 2019-01-10 with total page 337 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is an introduction to point set topology for undergraduates. Many of the classic textbooks on the subject cover the subject exhaustively and at the highest possible level of generality. The result of using traditional textbooks has been that students spend 2 semesters learning far more general topology on abstract spaces then most of them will ever need to use or know. More importantly, students get the impression from geometers and topologists in later courses that they "wasted" a year of their studies learning material that most mathematicians don't even consider topology anymore. This leaves many of them feeling deceived and frustrated. Unfortunately, the reaction has been in recent decades to write elementary topology textbooks that only present the barest minimum of point set topology needed for students in advanced geometry or algebraic topology. Indeed-some recent beginning textbooks in topology largely skip general topology altogether and jump straight into algebraic and geometric topology such as homotopy, curves and surfaces! We believe this ludicrous solution is essentially throwing the baby out with the bathwater. This reissued edition of Hall/ Spencer should seriously be considered by mathematicians as the benchmark for such a course. The book contains what we believe to be approximately the irreducible minimum of point set topology any student of mathematics needs to learn regardless of level or interest. The book is quite detailed, covering sufficient general topology of interest and use for analysts, geometers and topologists. The book falls into two rather distinct parts. The first half is concerned with an introductory study of topological and metric spaces. The basic operations with sets are introduced in Chapter I, relations and mappings are discussed, and an introduction to infinite and uncountable sets is given. Chapter 2 introduces the basic topological structure of the real numbers in a review of basic analysis. In Chapter 3, general topological and metric spaces are introduced and such topics as compactness, separation and continuous functions are discussed. Metric spaces are pursued further in Chapter 4, with discussions of local connectivity, countability, metrizability and completion being included. The second part is less elementary in character. The long Chapter 5 is concerned with giving topological characterizations of arcs, simple closed curves, and simple closed surfaces. Peano spaces are discussed and the Jordan curve theorem and Jordan-Schoenflies theorem are proved. Chapter 6 discusses partitionable spaces, a topic often missing from modern texts. Finally, Chapter 7 discusses the axiom of choice, Zorn's lemma (in the form commonly called the Hausdorff niaximality principle) and the Tychonoff product theorem. The book in particular will help students understand the deep connection between general topology and real and complex analysis. The most natural path towards understanding abstract topological spaces, general continuous mappings and topological invariants on families of open sets is to see how they directly generalize the usual structures of analysis on the real line. Also. Blue Collar Scholar founder/editor Karo Maestro has added his usual personal touch to the new edition, with a new preface on his own reflections on point set topology and recommendations for supplementary or subsequent study. The prerequisites for the text are very minimal-just calculus and some experience with rigorous proofs. This wonderful lost text in this new inexpensive edition will serve a new generation of mathematics students who need to learn this crucial foundational subject with a presentation that's both detailed and informative without being exhaustive. It will indoctrinate students into the beauty and simplicity of point-set topology and convince them of its' intrinsic importance-primarily to analysis, but also to other areas of mathematics.
In addition to serving as an introduction to the basics of point-set topology, this text bridges the gap between the elementary calculus sequence and higher-level mathematics courses. The versatile, original approach focuses on learning to read and write proofs rather than covering advanced topics. Based on lecture notes that were developed over many years at The University of Seattle, the treatment is geared toward undergraduate math majors and suitable for a variety of introductory courses. Starting with elementary concepts in logic and basic techniques of proof writing, the text defines topological and metric spaces and surveys continuity and homeomorphism. Additional subjects include product spaces, connectedness, and compactness. The final chapter illustrates topology's use in other branches of mathematics with proofs of the fundamental theorem of algebra and of Picard's existence theorem for differential equations. "This is a back-to-basics introductory text in point-set topology that can double as a transition to proofs course. The writing is very clear, not too concise or too wordy. Each section of the book ends with a large number of exercises. The optional first chapter covers set theory and proof methods; if the students already know this material you can start with Chapter 2 to present a straight topology course, otherwise the book can be used as an introduction to proofs course also." — Mathematical Association of America
Book Synopsis Elementary Point-Set Topology by : Andre L. Yandl
Download or read book Elementary Point-Set Topology written by Andre L. Yandl and published by Courier Dover Publications. This book was released on 2016-04-10 with total page 256 pages. Available in PDF, EPUB and Kindle. Book excerpt: In addition to serving as an introduction to the basics of point-set topology, this text bridges the gap between the elementary calculus sequence and higher-level mathematics courses. The versatile, original approach focuses on learning to read and write proofs rather than covering advanced topics. Based on lecture notes that were developed over many years at The University of Seattle, the treatment is geared toward undergraduate math majors and suitable for a variety of introductory courses. Starting with elementary concepts in logic and basic techniques of proof writing, the text defines topological and metric spaces and surveys continuity and homeomorphism. Additional subjects include product spaces, connectedness, and compactness. The final chapter illustrates topology's use in other branches of mathematics with proofs of the fundamental theorem of algebra and of Picard's existence theorem for differential equations. "This is a back-to-basics introductory text in point-set topology that can double as a transition to proofs course. The writing is very clear, not too concise or too wordy. Each section of the book ends with a large number of exercises. The optional first chapter covers set theory and proof methods; if the students already know this material you can start with Chapter 2 to present a straight topology course, otherwise the book can be used as an introduction to proofs course also." — Mathematical Association of America
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.
Book Synopsis Topology by : Tai-Danae Bradley
Download or read book Topology written by Tai-Danae Bradley and published by MIT Press. This book was released on 2020-08-18 with total page 167 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Students must prove all of the theorems in this undergraduate-level text, which features extensive outlines to assist in study and comprehension. Thorough and well-written, the treatment provides sufficient material for a one-year undergraduate course. The logical presentation anticipates students' questions, and complete definitions and expositions of topics relate new concepts to previously discussed subjects. Most of the material focuses on point-set topology with the exception of the last chapter. Topics include sets and functions, infinite sets and transfinite numbers, topological spaces and basic concepts, product spaces, connectivity, and compactness. Additional subjects include separation axioms, complete spaces, and homotopy and the fundamental group. Numerous hints and figures illuminate the text. Dover (2014) republication of the edition originally published by The Williams & Wilkins Company, Baltimore, 1975. See every Dover book in print at www.doverpublications.com
Book Synopsis A First Course in Topology by : Robert A Conover
Download or read book A First Course in Topology written by Robert A Conover and published by Courier Corporation. This book was released on 2014-05-21 with total page 276 pages. Available in PDF, EPUB and Kindle. Book excerpt: Students must prove all of the theorems in this undergraduate-level text, which features extensive outlines to assist in study and comprehension. Thorough and well-written, the treatment provides sufficient material for a one-year undergraduate course. The logical presentation anticipates students' questions, and complete definitions and expositions of topics relate new concepts to previously discussed subjects. Most of the material focuses on point-set topology with the exception of the last chapter. Topics include sets and functions, infinite sets and transfinite numbers, topological spaces and basic concepts, product spaces, connectivity, and compactness. Additional subjects include separation axioms, complete spaces, and homotopy and the fundamental group. Numerous hints and figures illuminate the text. Dover (2014) republication of the edition originally published by The Williams & Wilkins Company, Baltimore, 1975. See every Dover book in print at www.doverpublications.com
The essentials of point-set topology, complete with motivation and numerous examples Topology: Point-Set and Geometric presents an introduction to topology that begins with the axiomatic definition of a topology on a set, rather than starting with metric spaces or the topology of subsets of Rn. This approach includes many more examples, allowing students to develop more sophisticated intuition and enabling them to learn how to write precise proofs in a brand-new context, which is an invaluable experience for math majors. Along with the standard point-set topology topics—connected and path-connected spaces, compact spaces, separation axioms, and metric spaces—Topology covers the construction of spaces from other spaces, including products and quotient spaces. This innovative text culminates with topics from geometric and algebraic topology (the Classification Theorem for Surfaces and the fundamental group), which provide instructors with the opportunity to choose which "capstone" best suits his or her students. Topology: Point-Set and Geometric features: A short introduction in each chapter designed to motivate the ideas and place them into an appropriate context Sections with exercise sets ranging in difficulty from easy to fairly challenging Exercises that are very creative in their approaches and work well in a classroom setting A supplemental Web site that contains complete and colorful illustrations of certain objects, several learning modules illustrating complicated topics, and animations of particularly complex proofs
Book Synopsis Topology by : Paul L. Shick
Download or read book Topology written by Paul L. Shick and published by John Wiley & Sons. This book was released on 2011-10-14 with total page 291 pages. Available in PDF, EPUB and Kindle. Book excerpt: The essentials of point-set topology, complete with motivation and numerous examples Topology: Point-Set and Geometric presents an introduction to topology that begins with the axiomatic definition of a topology on a set, rather than starting with metric spaces or the topology of subsets of Rn. This approach includes many more examples, allowing students to develop more sophisticated intuition and enabling them to learn how to write precise proofs in a brand-new context, which is an invaluable experience for math majors. Along with the standard point-set topology topics—connected and path-connected spaces, compact spaces, separation axioms, and metric spaces—Topology covers the construction of spaces from other spaces, including products and quotient spaces. This innovative text culminates with topics from geometric and algebraic topology (the Classification Theorem for Surfaces and the fundamental group), which provide instructors with the opportunity to choose which "capstone" best suits his or her students. Topology: Point-Set and Geometric features: A short introduction in each chapter designed to motivate the ideas and place them into an appropriate context Sections with exercise sets ranging in difficulty from easy to fairly challenging Exercises that are very creative in their approaches and work well in a classroom setting A supplemental Web site that contains complete and colorful illustrations of certain objects, several learning modules illustrating complicated topics, and animations of particularly complex proofs
This text explains nontrivial applications of metric space topology to analysis. Covers metric space, point-set topology, and algebraic topology. Includes exercises, selected answers, and 51 illustrations. 1983 edition.
Book Synopsis Introduction to Topology by : Theodore W. Gamelin
Download or read book Introduction to Topology written by Theodore W. Gamelin and published by Courier Corporation. This book was released on 2013-04-22 with total page 258 pages. Available in PDF, EPUB and Kindle. Book excerpt: This text explains nontrivial applications of metric space topology to analysis. Covers metric space, point-set topology, and algebraic topology. Includes exercises, selected answers, and 51 illustrations. 1983 edition.
This book provides a concise introduction to topology and is necessary for courses in differential geometry, functional analysis, algebraic topology, etc. Topology is a fundamental tool in most branches of pure mathematics and is also omnipresent in more applied parts of mathematics. Therefore students will need fundamental topological notions already at an early stage in their bachelor programs. While there are already many excellent monographs on general topology, most of them are too large for a first bachelor course. Topology fills this gap and can be either used for self-study or as the basis of a topology course.
Book Synopsis Topology by : Stefan Waldmann
Download or read book Topology written by Stefan Waldmann and published by Springer. This book was released on 2014-08-05 with total page 143 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a concise introduction to topology and is necessary for courses in differential geometry, functional analysis, algebraic topology, etc. Topology is a fundamental tool in most branches of pure mathematics and is also omnipresent in more applied parts of mathematics. Therefore students will need fundamental topological notions already at an early stage in their bachelor programs. While there are already many excellent monographs on general topology, most of them are too large for a first bachelor course. Topology fills this gap and can be either used for self-study or as the basis of a topology course.