One of the most important milestones in mathematics in the twentieth century was the development of topology as an independent field of study and the subsequent systematic application of topological ideas to other fields of mathematics.
While there are many other works on introductory topology, this volume employs a methodology somewhat different from other texts. Metric space and point-set topology material is treated in the first two chapters; algebraic topological material in the remaining two. The authors lead readers through a number of nontrivial applications of metric space topology to analysis, clearly establishing the relevance of topology to analysis. Second, the treatment of topics from elementary algebraic topology concentrates on results with concrete geometric meaning and presents relatively little algebraic formalism; at the same time, this treatment provides proof of some highly nontrivial results. By presenting homotopy theory without considering homology theory, important applications become immediately evident without the necessity of a large formal program.
Prerequisites are familiarity with real numbers and some basic set theory. Carefully chosen exercises are integrated into the text (the authors have provided solutions to selected exercises for the Dover edition), while a list of notations and bibliographical references appear at the end of the book.

According to the authors of this highly useful compendium, focusing on examples is an extremely effective method of involving undergraduate mathematics students in actual research. It is only as a result of pursuing the details of each example that students experience a significant increment in topological understanding. With that in mind, Professors Steen and Seebach have assembled 143 examples in this book, providing innumerable concrete illustrations of definitions, theorems, and general methods of proof. Far from presenting all relevant examples, however, the book instead provides a fruitful context in which to ask new questions and seek new answers.
Ranging from the familiar to the obscure, the examples are preceded by a succinct exposition of general topology and basic terminology and theory. Each example is treated as a whole, with a highly geometric exposition that helps readers comprehend the material. Over 25 Venn diagrams and reference charts summarize the properties of the examples and allow students to scan quickly for examples with prescribed properties. In addition, discussions of general methods of constructing and changing examples acquaint readers with the art of constructing counterexamples. The authors have included an extensive collection of problems and exercises, all correlated with various examples, and a bibliography of 140 sources, tracing each uncommon example to its origin.
This revised and expanded second edition will be especially useful as a course supplement and reference work for students of general topology. Moreover, it gives the instructor the flexibility to design his own course while providing students with a wealth of historically and mathematically significant examples. 1978 edition.

Differential geometry and topology are essential tools for many theoretical physicists, particularly in the study of condensed matter physics, gravity, and particle physics. Written by physicists for physics students, this text introduces geometrical and topological methods in theoretical physics and applied mathematics. It assumes no detailed background in topology or geometry, and it emphasizes physical motivations, enabling students to apply the techniques to their physics formulas and research.
Thoroughly recommended by The Physics Bulletin, this volume's physics applications range from condensed matter physics and statistical mechanics to elementary particle theory. Its main mathematical topics include differential forms, homotopy, homology, cohomology, fiber bundles, connection and covariant derivatives, and Morse theory.

In most major universities one of the three or four basic first-year graduate mathematics courses is algebraic topology. This introductory text is suitable for use in a course on the subject or for self-study, featuring broad coverage and a readable exposition, with many examples and exercises. The four main chapters present the basics: fundamental group and covering spaces, homology and cohomology, higher homotopy groups, and homotopy theory generally. The author emphasizes the geometric aspects of the subject, which helps students gain intuition. A unique feature is the inclusion of many optional topics not usually part of a first course due to time constraints: Bockstein and transfer homomorphisms, direct and inverse limits, H-spaces and Hopf algebras, the Brown representability theorem, the James reduced product, the Dold-Thom theorem, and Steenrod squares and powers.

How is a subway map different from other maps? What makes a knot knotted? What makes the Möbius strip one-sided? These are questions of topology, the mathematical study of properties preserved by twisting or stretching objects. In the 20th century topology became as broad and fundamental as algebra and geometry, with important implications for science, especially physics.

In this Very Short Introduction Richard Earl gives a sense of the more visual elements of topology (looking at surfaces) as well as covering the formal definition of continuity. Considering some of the eye-opening examples that led mathematicians to recognize a need for studying topology, he pays homage to the historical people, problems, and surprises that have propelled the growth of this field.

ABOUT THE SERIES: The Very Short Introductions series from Oxford University Press contains hundreds of titles in almost every subject area. These pocket-sized books are the perfect way to get ahead in a new subject quickly. Our expert authors combine facts, analysis, perspective, new ideas, and enthusiasm to make interesting and challenging topics highly readable.

The finite generation theorem is a major achievement in modern algebraic geometry. Based on the minimal model theory, it states that the canonical ring of an algebraic variety defined over a field of characteristic 0 is a finitely generated graded ring. This graduate-level text is the first to explain this proof. It covers the progress on the minimal model theory over the last 30 years, culminating in the landmark paper on finite generation by Birkar‒Cascini‒Hacon‒McKernan. Building up to this proof, the author presents important results and techniques that are now part of the standard toolbox of birational geometry, including Mori's bend-and-break method, vanishing theorems, positivity theorems, and Siu's analysis on multiplier ideal sheaves. Assuming only the basics in algebraic geometry, the text keeps prerequisites to a minimum with self-contained explanations of terminology and theorems.

Highly regarded for its exceptional clarity, imaginative and instructive exercises, and fine writing style, this concise book offers an ideal introduction to the fundamentals of topology. Originally conceived as a text for a one-semester course, it is directed to undergraduate students whose studies of calculus sequence have included definitions and proofs of theorems. The book's principal aim is to provide a simple, thorough survey of elementary topics in the study of collections of objects, or sets, that possess a mathematical structure.

The author begins with an informal discussion of set theory in Chapter 1, reserving coverage of countability for Chapter 5, where it appears in the context of compactness. In the second chapter Professor Mendelson discusses metric spaces, paying particular attention to various distance functions which may be defined on Euclidean n-space and which lead to the ordinary topology.

Chapter 3 takes up the concept of topological space, presenting it as a generalization of the concept of a metric space. Chapters 4 and 5 are devoted to a discussion of the two most important topological properties: connectedness and compactness. Throughout the text, Dr. Mendelson, a former Professor of Mathematics at Smith College, has included many challenging and stimulating exercises to help students develop a solid grasp of the material presented.

This excellent introduction to topology eases first-year math students and general readers into the subject by surveying its concepts in a descriptive and intuitive way, attempting to build a bridge from the familiar concepts of geometry to the formalized study of topology. The first three chapters focus on congruence classes defined by transformations in real Euclidean space. As the number of permitted transformations increases, these classes become larger, and their common topological properties become intuitively clear. Chapters 4-12 give a largely intuitive presentation of selected topics. In the remaining five chapters, the author moves to a more conventional presentation of continuity, sets, functions, metric spaces, and topological spaces. Exercises and Problems. 101 black-and-white illustrations. 1974 edition.

Classroom-tested and much-cited, this concise text is designed for undergraduates. It offers a valuable and instructive introduction to the basic concepts of topology, taking an intuitive rather than an axiomatic viewpoint. 1962 edition.

Hailed by the Bulletin of the American Mathematical Society as a very welcome addition to the mathematical literature, this text is appropriate for advanced undergraduates and graduate students. Written by two internationally renowned mathematicians, its accessible treatment requires no previous knowledge of algebraic topology.
Starting with basic definitions of knots and knot types, the text proceeds to examinations of fundamental and free groups. A survey of the historic foundation for the notion of group presentation is followed by a careful proof of the theorem of Tietze and several examples of its use. Subsequent chapters explore the calculation of fundamental groups, the presentation of a knot group, the free calculus and the elementary ideals, and the knot polynomials and their characteristic properties. The text concludes with three helpful appendixes and a guide to the literature.

Among the best available reference introductions to general topology, this volume is appropriate for advanced undergraduate and beginning graduate students. Its treatment encompasses two broad areas of topology: continuous topology, represented by sections on convergence, compactness, metrization and complete metric spaces, uniform spaces, and function spaces; and geometric topology, covered by nine sections on connectivity properties, topological characterization theorems, and homotopy theory. Many standard spaces are introduced in the related problems that accompany each section (340 exercises in all). The text's value as a reference work is enhanced by a collection of historical notes, a bibliography, and index. 1970 edition. 27 figures.

Written by the founder of functional analysis, this is the first text on linear operator theory. Additional topics include the calculus of variations and theory of integral equations. 1987 edition.

Manifolds, the higher-dimensional analogs of smooth curves and surfaces, are fundamental objects in modern mathematics. Combining aspects of algebra, topology, and analysis, manifolds have also been applied to classical mechanics, general relativity, and quantum field theory. In this streamlined introduction to the subject, the theory of manifolds is presented with the aim of helping the reader achieve a rapid mastery of the essential topics. By the end of the book the reader should be able to compute, at least for simple spaces, one of the most basic topological invariants of a manifold, its de Rham cohomology. Along the way, the reader acquires the knowledge and skills necessary for further study of geometry and topology. The requisite point-set topology is included in an appendix of twenty pages; other appendices review facts from real analysis and linear algebra. Hints and solutions are provided to many of the exercises and problems. This work may be used as the text for a one-semester graduate or advanced undergraduate course, as well as by students engaged in self-study. Requiring only minimal undergraduate prerequisites, 'Introduction to Manifolds' is also an excellent foundation for Springer's GTM 82, 'Differential Forms in Algebraic Topology'.

Cohomology operations are at the center of a major area of activity in algebraic topology. This technique for supplementing and enriching the algebraic structure of the cohomology ring has been instrumental to important progress in general homotopy theory and in specific geometric applications. For both theoretical and practical reasons, the formal properties of families of operations have received extensive analysis.
This text focuses on the single most important sort of operations, the Steenrod squares. It constructs these operations, proves their major properties, and provides numerous applications, including several different techniques of homotopy theory useful for computation. In the later chapters, the authors place special emphasis on calculations in the stable range. The text provides an introduction to methods of Serre, Toda, and Adams, and carries out some detailed computations. Prerequisites include a solid background in cohomology theory and some acquaintance with homotopy groups.

Algebraic topology is a basic part of modern mathematics, and some knowledge of this area is indispensable for any advanced work relating to geometry, including topology itself, differential geometry, algebraic geometry, and Lie groups. This book provides a detailed treatment of algebraic topology both for teachers of the subject and for advanced graduate students in mathematics either specializing in this area or continuing on to other fields.

J. Peter May's approach reflects the enormous internal developments within algebraic topology over the past several decades, most of which are largely unknown to mathematicians in other fields. But he also retains the classical presentations of various topics where appropriate. Most chapters end with problems that further explore and refine the concepts presented. The final four chapters provide sketches of substantial areas of algebraic topology that are normally omitted from introductory texts, and the book concludes with a list of suggested readings for those interested in delving further into the field.

The ideal review for your general topology course

More than 40 million students have trusted Schaum's Outlines for their expert knowledge and helpful solved problems. Written by renowned experts in their respective fields, Schaum's Outlines cover everything from math to science, nursing to language. The main feature for all these books is the solved problems. Step-by-step, authors walk readers through coming up with solutions to exercises in their topic of choice.

• 391 solved problems
• 356 supplementary problems
• Teaches effective problem-solving
• Outline format supplies a concise guide to the standard college courses in General Topology
• Supports and supplements the leading General Topology textbooks
• Detailed explanations and practice problems in general topology
• Comprehensive review of specialized topics in topology

This is the third edition of a classic text, previously published in 1968, 1988, and now extended, revised, retitled, updated, and reasonably priced. Throughout it gives motivation and context for theorems and definitions. Thus the definition of a topology is first related to the example of the real line; it is then given in terms of the intuitive notion of neighbourhoods, and then shown to be equivalent to the elegant but spare definition in terms of open sets. Many constructions of topologies are shown to be necessitated by the desire to construct continuous functions, either from or into a space. This is in the modern categorical spirit, and often leads to clearer and simpler proofs. There is a full treatment of finite cell complexes, with the cell decompositions given of projective spaces, in the real, complex and quaternionic cases. This is based on an exposition of identification spaces and adjunction spaces. The exposition of general topology ends with a description of the topology for function spaces, using the modern treatment of the test-open topology, from compact Hausdorff spaces, and so a description of a convenient category of spaces (a term due to the author) in the non Hausdorff case. The second half of the book demonstrates how the use of groupoids rather than just groups gives in 1-dimensional homotopy theory more powerful theorems with simpler proofs. Some of the proofs of results on the fundamental groupoid would be difficult to envisage except in the form given: We verify the required universal property'. This is in the modern categorical spirit. Chapter 6 contains the development of the fundamental groupoid on a set of base points, including the background in category theory. The proof of the van Kampen Theorem in this general form resolves a failure of traditional treatments, in giving a direct computation of the fundamental group of the circle, as well as more complicated examples. Chapter 7 uses the notion of cofibration to develop the notion of operations of the fundamental groupoid on certain sets of homotopy classes. This allows for an important theorem on gluing homotopy equivalences by a method which gives control of the homotopies involved. This theorem first appeared in the 1968 edition. Also given is the family of exact sequences arising from a fibration of groupoids. The development of Combinatorial Groupoid Theory in Chapter 8 allows for unified treatments of free groups, free products of groups, and HNN-extensions, in terms of pushouts of groupoids, and well models the topology of gluing spaces together. These methods lead in Chapter 9 to results on the Phragmen-Brouwer Property, with a Corollary that the complement of any arc in an n-sphere is connected, and then to a proof of the Jordan Curve Theorem. Chapter 10 on covering spaces is again fully in the base point free spirit; it proves the natural theorem that for suitable spaces X, the category of covering spaces of X is equivalent to the category of covering morphisms of the fundamental groupoid of X. This approach gives a convenient way of obtaining covering maps from covering morphisms, and leads easily to traditional results using operations of the fundamental group. Results on pullbacks of coverings are proved using a Mayer-Vietoris type sequence. No other text treats the whole theory directly in this way. Chapter 11 is on Orbit Spaces and Orbit Groupoids, and gives conditions for the fundamental groupoid of the orbit space to be the orbit groupoid of the fundamental groupoid. No other topology text treats this important area. Comments on the relations to the literature are given in Notes at the end of each Chapter. There are over 500 exercises, 114 figures, numerous diagrams. See http: //www.bangor.ac.uk/r.brown/topgpds.html for more information. See http: //mathdl.maa.org/mathDL/19/?rpa=reviews&sa=viewBook& bookId=69421 for a Mathematical Association of America review.

This well-written and engaging volume, intended for undergraduates, introduces knot theory, an area of growing interest in contemporary mathematics. The hands-on approach features many exercises to be completed by readers. Prerequisites are only a basic familiarity with linear algebra and a willingness to explore the subject in a hands-on manner.
The opening chapter offers activities that explore the world of knots and links -- including games with knots -- and invites the reader to generate their own questions in knot theory. Subsequent chapters guide the reader to discover the formal definition of a knot, families of knots and links, and various knot notations. Additional topics include combinatorial knot invariants, knot polynomials, unknotting operations, and virtual knots.

The aim of the textbook is two-fold: first to serve as an introductory graduate course in Algebraic Topology and then to provide an application-oriented presentation of some fundamental concepts in Algebraic Topology to the fixed point theory.

A simple approach based on point-set Topology is used throughout to introduce many standard constructions of fundamental and homological groups of surfaces and topological spaces. The approach does not rely on Homological Algebra. The constructions of some spaces using the quotient spaces such as the join, the suspension, and the adjunction spaces are developed in the setting of Topology only.

The computations of the fundamental and homological groups of many surfaces and topological spaces occupy large parts of the book (sphere, torus, projective space, Mobius band, Klein bottle, manifolds, adjunctions spaces). Borsuk's theory of retracts which is intimately related to the problem of the extendability of continuous functions is developed in details. This theory together with the homotopy theory, the lifting and covering maps may serve as additional course material for students involved in General Topology.

The book comprises 280 detailed worked examples, 320 exercises (with hints or references), 80 illustrative figures, and more than 80 commutative diagrams to make it more oriented towards applications (maps between spheres, Borsuk-Ulam Theory, Fixed Point Theorems, ...) As applications, the book offers some existence results on the solvability of some nonlinear differential equations subject to initial or boundary conditions.

The book is suitable for students primarily enrolled in Algebraic Topology, General Topology, Homological Algebra, Differential Topology, Differential Geometry, and Topological Geometry. It is also useful for advanced undergraduate students who aspire to grasp easily some new concepts in Algebraic Topology and Applications. The textbook is practical both as a teaching and research document for Bachelor, Master students, and first-year PhD students since it is accessible to any reader with a modest understanding of topological spaces.

The book aspires to fill a gap in the existing literature by providing a research and teaching document which investigates both the theory and the applications of Algebraic Topology in an accessible way without missing the main results of the topics covered.

This sixth entry in the highly acclaimed Math Girls series focuses on the Poincaré Conjecture, a fundamental problem in topology first proposed in 1904. While the problem is simply stated and easily understood, it resisted proof throughout the twentieth century. Russian mathematician Grigori Perelman finally completed that effort, publishing a series of papers in 2002 that provided missing details for an argument that includes a solution. In this book, you will join Miruka and friends as they learn about topology from its very beginnings: the Seven Bridges of Königsberg problem that Leonhard Euler investigated in 1736. After that you will learn about interesting objects like the Möbius strip and the Klein bottle, topological spaces and continuous mappings, homeomophism and homotopy, and non-Euclidean geometries. Along the way, you will also learn about differential equations, Fourier series, the heat equation, and a trigonometric training regimen. The book concludes with an introduction to Hamilton's Ricci flow, a crucial tool in Perelman's work on the Poincaré Conjecture. Math Girls 6: The Poincaré Conjecture has something for anyone interested in mathematics, from advanced high school to college students and educators.