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• 1. Basics
• 2. Complexes and homology groups
• 3. Topological persistence
• 4. General persistence
• 5. Generators and optimality
• 6. Topological analysis of point clouds
• 7. Reeb graphs
• 8. Topological analysis of graphs
• 9. Cover, nerve and Mapper
• 10. Discrete Morse theory and applications
• 11. Multiparameter persistence and decomposition
• 12. Multiparameter persistence and distances
• 13. Topological persistence and machine learning.
• (source: Nielsen Book Data)

Topological data analysis (TDA) has emerged recently as a viable tool for analyzing complex data, and the area has grown substantially both in its methodologies and applicability. Providing a computational and algorithmic foundation for techniques in TDA, this comprehensive, self-contained text introduces students and researchers in mathematics and computer science to the current state of the field. The book features a description of mathematical objects and constructs behind recent advances, the algorithms involved, computational considerations, as well as examples of topological structures or ideas that can be used in applications. It provides a thorough treatment of persistent homology together with various extensions - like zigzag persistence and multiparameter persistence - and their applications to different types of data, like point clouds, triangulations, or graph data. Other important topics covered include discrete Morse theory, the Mapper structure, optimal generating cycles, as well as recent advances in embedding TDA within machine learning frameworks.
(source: Nielsen Book Data)

• Preface
• Introduction
• Topological Spaces
• Continuous Maps and Homeomorphisms
• Connectedness
• Separation Axioms and Quotient Axioms
• Compactness
• Product Spaces and Quotient Spaces
• Appendix: Some Elementary Inequalities--.
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This book is intended as a one-semester course in general topology, a.k.a. point-set topology, for undergraduate students as well as first-year graduate students. Such a course is considered a prerequisite for further studying analysis, geometry, manifolds, and certainly, for a career of mathematical research. Researchers may find it helpful especially from the comprehensive indices.General topology resembles a language in modern mathematics. Because of this, the book is with a concentration on basic concepts in general topology, and the presentation is of a brief style, both concise and precise. Though it is hard to determine exactly which concepts therein are basic and which are not, the author makes efforts in the selection according to personal experience on the occurrence frequency of notions in advanced mathematics, and to related books that have received admirable reviews.This book also contains exercises for each chapter with selected solutions. Interrelationships among concepts are taken into account frequently. Twelve particular topological spaces are repeatedly exploited, which serve as examples to learn new concepts based on old ones.
(source: Nielsen Book Data)

• Chapter 1. Topological Spaces.-
• Chapter 2. Continuity and Products.-
• Chapter 3. Connectedness.-
• Chapter 4. Convergence.-
• Chapter 5. Countability axioms.-
• Chapter 6. Compactness.-
• Chapter 7. Topological Constructions.-
• Chapter 8. Separation Axioms.-
• Chapter 9. Paracompactness and Metrisability.-
• Chapter 10. Completeness.-
• Chapter 11. Function Spaces.-
• Chapter 12. Topological Groups.-
• Chapter 13. Transformation Groups.-
• Chapter 14. The fundamental Group.-
• Chapter 15. Covering Spaces.
• (source: Nielsen Book Data)

Topology is a large subject with several branches, broadly categorized as algebraic topology, point-set topology, and geometric topology. Point-set topology is the main language for a broad range of mathematical disciplines, while algebraic topology offers as a powerful tool for studying problems in geometry and numerous other areas of mathematics. This book presents the basic concepts of topology, including virtually all of the traditional topics in point-set topology, as well as elementary topics in algebraic topology such as fundamental groups and covering spaces. It also discusses topological groups and transformation groups. When combined with a working knowledge of analysis and algebra, this book offers a valuable resource for advanced undergraduate and beginning graduate students of mathematics specializing in algebraic topology and harmonic analysis.
(source: Nielsen Book Data)

• 1: What is Topology?
• 2: Making Surfaces
• 3: Thinking Continuously
• 4: The Plane and Other Spaces
• 5: Flavours of Topology
• 6: More on Surfaces
• 7: Knot to Be Historical Timeline Further Reading Index.
• (source: Nielsen Book Data)

How is a subway map different from other maps? What makes a knot knotted? What makes the Moebius 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.
(source: Nielsen Book Data)

• 1.Introduction.-
• 2. Prerequisites.-
• 3. Products of Relations.-
• 4. Meet and Join as Relations.-
• 5. Applying Relations in Topology.-
• 6. Construction of Topologies.-
• 7. Closures and their Aumann Contacts.-
• 8. Proximity and Nearness.-
• 9. Frames.-
• 10. Simplicial Complexes.
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This book introduces and develops new algebraic methods to work with relations, often conceived as Boolean matrices, and applies them to topology. Although these objects mirror the matrices that appear throughout mathematics, numerics, statistics, engineering, and elsewhere, the methods used to work with them are much less well known. In addition to their purely topological applications, the volume also details how the techniques may be successfully applied to spatial reasoning and to logics of computer science. Topologists will find several familiar concepts presented in a concise and algebraically manipulable form which is far more condensed than usual, but visualized via represented relations and thus readily graspable. This approach also offers the possibility of handling topological problems using proof assistants.
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• PrefaceI Euclidean Topolog
• y1. Introduction to Topology1.1 Deformations1.2 Topological Space
• s2. Metric Topology in Euclidean Space2.1 Distance2.2 Continuity and Homeomorphism2.3 Compactness and Limits2.4 Connectedness2.5 Metric Spaces in Genera
• l3. Vector Fields in the Plane3.1 Trajectories and Phase Portraits3.2 Index of a Critical Point3.3 *Nullclines and Trapping RegionsII Abstract Topology with Application
• s4. Abstract Point-Set Topology4.1 The Definition of a Topology4.2 Continuity and Limits4.3 Subspace Topology and Quotient Topology4.4 Compactness and Connectedness4.5 Product and Function Spaces4.6 *The Infinitude of the Prime
• s5. Surfaces5.1 Surfaces and Surfaces-with-Boundary5.2 Plane Models and Words5.3 Orientability5.4 Euler Characteristi
• c6. Applications in Graphs and Knots6.1 Graphs and Embeddings6.2 Graphs, Maps, and Coloring Problems6.3 Knots and Links6.4 Knot ClassificationIII Basic Algebraic Topolog
• y7. The Fundamental Group7.1 Algebra of Loops7.2 Fundamental Group as Topological Invariant7.3 Covering Spaces and the Circle7.4 Compact Surfaces and Knot Complements7.5 *Higher Homotopy Group
• s8. Introduction to Homology8.1 Rational Homology8.2 Integral HomologyAppendixesA. Review of Set Theory and FunctionsA.1 Sets and Operations on SetsA.2 Relations and FunctionsB. Group Theory and Linear AlgebraB.1 GroupsB.2 Linear AlgebraC. Selected SolutionsD. NotationsBibliographyIndex.
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Topology-the branch of mathematics that studies the properties of spaces that remain unaffected by stretching and other distortions-can present significant challenges for undergraduate students of mathematics and the sciences. Understanding Topology aims to change that.The perfect introductory topology textbook, Understanding Topology requires only a knowledge of calculus and a general familiarity with set theory and logic. Equally approachable and rigorous, the book's clear organization, worked examples, and concise writing style support a thorough understanding of basic topological principles. Professor Shaun V. Ault's unique emphasis on fascinating applications, from mapping DNA to determining the shape of the universe, will engage students in a way traditional topology textbooks do not.This groundbreaking new text:* presents Euclidean, abstract, and basic algebraic topology* explains metric topology, vector spaces and dynamics, point-set topology, surfaces, knot theory, graphs and map coloring, the fundamental group, and homology* includes worked example problems, solutions, and optional advanced sections for independent projectsFollowing a path that will work with any standard syllabus, the book is arranged to help students reach that "Aha!" moment, encouraging readers to use their intuition through local-to-global analysis and emphasizing topological invariants to lay the groundwork for algebraic topology.
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• 1 Introduction 1.1 Scope of the Thesis 1.2 Outline of the Thesis 1.3 Quantum Hall States 1.4 Topological Insulators 1.5 Weyl and Dirac Semimetals 1.6 -(BEDT-TTF)2I3 1.7 Topological Mott Insulators 1.8 Topological Crystalline Insulators 1.9 Classification of Topological States of Matter 2 Interacting Dirac Fermions in (3+1) Dimensions 2.1 Model 2.2 Renormalization Group Analysis 2.3 Density of States 2.4 Electromagnetic Properties 2.5 Spectral Function 2.6 Electric Conductivity 2.7 Energy Gap 2.8 Discussions and Summary 3 Tilted Dirac Cones in Two Dimensions 3.1 Model 3.2 Perturbative Renormalization Group Analysis 3.3 Spin Susceptibility 3.4 Discussions and Summary 4 Generalized Hund's Rule for Two-Atom Systems 4.1 Model 4.2 Results 4.3 Perturbative Calculation 4.4 Entanglement Entropy 4.5 Symmetry 4.6 Discussions and Summary 5 Interacting Topological Crystalline Insulators 5.1 Classification in Two Dimensions 5.2 Interacting TCIs in Three Dimensions 5.3 Discussions and Summary 6 Conclusions and Prospects.
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This thesis elucidates electron correlation effects in topological matter whose electronic states hold nontrivial topological properties robust against small perturbations. In addition to a comprehensive introduction to topological matter, this thesis provides a new perspective on correlated topological matter. The book comprises three subjects, in which electron correlations in different forms are considered. The first focuses on Coulomb interactions for massless Dirac fermions. Using a perturbative approach, the author reveals emergent Lorentz invariance in a low-energy limit and discusses how to probe the Lorentz invariance experimentally. The second subject aims to show a principle for synthesizing topological insulators with common, light elements. The interplay between the spin-orbit interaction and electron correlation is considered, and Hund's rule and electron filling are consequently found to play a key role for a strong spin-orbit interaction important for topological insulators. The last subject is classification of topological crystalline insulators in the presence of electron correlation. Unlike non-interacting topological insulators, such two- and three-dimensional correlated insulators with mirror symmetry are demonstrated to be characterized, respectively, by the Z4 and Z8 group by using the bosonization technique and a geometrical consideration.
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• Topological Homogeneity.- Some Recent Progress Concerning Topology of Fractals.- A biased view of topology as a tool in functional analysis.- Large scale versus small scale.- Descriptive aspects of Rosenthal compacta.- Minimality conditions in topological groups.- Set-Theoretic update on Topology.- Topics in Dimension Theory.- Representations of dynamical systems on Banach spaces.- Generalized metrizable spaces.- Permanence in Coarse Geometry.- Selections and Hyperspaces.- Continuum Theory.- Almost disjoint families and topology.- Some Topics in Geometric Topology II.- Topological aspects of dynamics of pairs, tuples and sets.- Continuous selections of multivalued mappings.- The combinatorics of open covers.- Covering properties.- Paratopological and semitopological groups vs topological groups.
• (source: Nielsen Book Data)

The book presents surveys describing recent developments in most of the primary subfields of General Topology, and its applications to Algebra and Analysis during the last decade, following the previous editions (North Holland, 1992 and 2002). The book was prepared in connection with the Prague Topological Symposium, held in 2011. During the last 10 years the focus in General Topology changed and therefore the selection of topics differs from that chosen in 2002. The following areas experienced significant developments: Fractals, Coarse Geometry/Topology, Dimension Theory, Set Theoretic Topology and Dynamical Systems.
(source: Nielsen Book Data)

• 1 Introduzione geometrica alla topologia
• 2 Insiemi
• 3 Strutture topologiche
• 4 Connessione e compattezza
• 5 Quozienti topologici
• 6 Successioni
• 7 Variet, prodotti infiniti e paracompattezza
• 8 Complementi di topologia generale
• 9 Intermezzo
• 10 Omotopia
• 11 Il gruppo fondamentale
• 12 Rivestimenti
• 13 Monodromia
• 14 Il teorema di Van Kampen
• 15 Complementi di topologia algebrica
• 16 Suggerimenti e soluzioni di alcuni esercizi.

Nato dall'esperienza dell'autore nell'insegnamento della topologia agli studenti del corso di Laurea in Matematica, questo libro contiene le nozioni fondamentali di topologia generale ed una introduzione alla topologia algebrica. La scelta degli argomenti, il loro ordine di presentazione e, soprattutto, il tipo di esposizione tiene conto delle tendenze attuali nell'insegnamento della topologia e delle novit nella struttura dei corsi di Laurea scientifici conseguenti all'introduzione del sistema 3+2. Questa seconda edizione, oltre a semplificare alcune dimostrazioni, presenta una sostanziale riscrittura della parte sui rivestimenti e l'aggiunta di ulteriori esempi; il numero complessivo di esercizi proposti stato portato a 500 ed il numero di quelli svolti a 120.

• Introduction
• Topological Spaces and Continuity
• Construction of Topological Spaces
• Convergence in Topological Spaces
• Compactness
• Continuous Functions
• Baire's Theorem.

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.

• Preface
• Part I. Fundamentals: 1.1. What is topology?
• 1.2. First definitions
• 1.3 Mappings
• 1.4. The separation axioms
• 1.5. Compactness
• 1.6. Homeomorphisms
• 1.7. Connectedness
• 1.8. Path-connectedness
• 1.9. Continua
• 1.10. Totally disconnected spaces
• 1.11. The Cantor set
• 1.12. Metric spaces
• 1.13. Metrizability
• 1.14. Baire's theorem
• 1.15. Lebesgue's lemma and Lebesgue numbers
• Part II. Advanced Properties: 2.1 Basis and subbasis
• 2.2. Product spaces
• 2.3. Relative topology
• 2.4. First countable and second countable
• 2.5. Compactifications
• 2.6. Quotient topologies
• 2.7. Uniformities
• 2.8. Morse theory
• 2.9. Proper mappings
• 2.10. Paracompactness
• Part III. Moore-Smith Convergence and Nets: 3.1. Introductory remarks
• 3.2. Nets
• Part IV. Function Spaces: 4.1. Preliminary ideas
• 4.2. The topology of pointwise convergence
• 4.3. The compact-open topology
• 4.4. Uniform convergence
• 4.5. Equicontinuity and the Ascoli-Arzela theorem
• 4.6. The Weierstrass approximation theorem
• Table of notation
• Glossary
• Bibliography
• Index.
• (source: Nielsen Book Data)

A Guide to Topology is an introduction to basic topology for graduate or advanced undergraduate students. It covers point-set topology, Moore-Smith convergence and function spaces. It treats continuity, compactness, the separation axioms, connectedness, completeness, the relative topology, the quotient topology, the product topology, and all the other fundamental ideas of the subject. The book is filled with examples and illustrations. Students studying for exams will find this book to be a concise, focused and informative resource. Professional mathematicians who need a quick review of the subject, or need a place to look up a key fact, will find this book to be a useful resource too.
(source: Nielsen Book Data)

• Homeomorphic Sets.- Topological Properties.- Equivalent Subsets.- Surfaces and Spaces.- Polyhedra.- Winding Number.
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Topologyhasbeenreferredtoas"rubber-sheetgeometry".Thenameisapt , for the subject is concerned with properties of an object that would be preserved, no matter how much it is stretched, squashed, or distorted, so long as it is not in any way torn apart or glued together. One's ?rst reaction might be that such animprecise-soundingsubjectcouldhardlybepartofseriousmathematics , and wouldbeunlikelytohaveapplicationsbeyondtheamusementofsimpleparlo ur games. This reaction could hardly be further from the truth. Topology is one of the most important and broad-ranging disciplines of modern mathematics. It is a subject of great precision and of breadth of development. It has vastly many applications, some of great importance, ranging from particle physics to cosmology, and from hydrodynamics to algebra and number theory. It is also a subject of great beauty and depth. To appreciate something of this, it is not necessary to delve into the more obscure aspects of mathematical formalism. For topology is, at least initially, a very visual subject. Some of its concepts apply to spaces of large numbers of dimensions, and therefore do not easily submit to reasoning that depends upon direct pictorial representation. But even in such cases, important insights can be obtained from the visual - rusal of a simple geometrical con?guration. Although much modern topology depends upon ?nely tuned abstract algebraic machinery of great mathematical sophistication, the underlying ideas are often very simple and can be appre- ated by the examination of properties of elementary-looking drawings.
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• Part 1. General Topology 1. Selected ordered space problems (H. Bennett and D. Lutzer) 2. Problems on star-covering properties (M. Bonanzinga and M. Matveev) 3. Function space topologies (D.N. Georgiou, S.D. Iliadis and F. Mynard) 4. Spaces and mappings: special networks (C. Liu and Y. Tanaka) 5. Extension problems of real-valued continuous functions (H. Ohta and K. Yamazaki) 6. LE(k)-spaces (O. Okunev) 7. Problems on (ir) resolvability (O. Pavlov) 8. Topological games and Ramsey theory (M. Scheepers) 9. Selection principles and special sets of reals (B. Tsaban)
• Part 2. Set-theoretic Topology 10. Introduction: Twenty problems in set-theoretic topology (M. Hrusak and J.T. Moore) 11. Thin-tall spaces and cardinal sequences (J. Bagaria) 12. Sequential order (A. Dow) 13. On D-spaces (T. Eisworth) 14. The fourth head of BN (I. Farah) 15. Are stratifiable spaces M1? (G. Gruenhage) 16. Perfect compacta and basis problems in topology (G. Gruenhage and J.T. Moore) 17. Selection problems for hyperspaces (V. Gutev and T. Nogura) 18. Efimov's problem (K.P. Hart) 19. Completely separable MAD families (M. Hrusak and P. Simon) 20. Good, splendid and Jakovlev (I. Juhasz and W.A.R. Weiss) 21. Homogeneous compacta (J. van Mill) 22. Compact spaces with hereditarily normal squares (J.T. Moore) 23. The metrization problem for Frechet groups (J.T. Moore and S. Todorcevic) 24. Cech-Stone remainders of discrete spaces (P.J. Nyikos) 25. First countable, countably compact, noncompact spaces (P.J. Nyikos) 26. Linearly Lindeloef problems (E. Pearl) 27. Small Dowker spaces (P.J. Szeptycki) 28. Reflection of topological properties to N1 (F.D. Tall) 29. The Scarborough-Stone problem (J.E. Vaughan)
• Part 3. Continuum Theory 30. Questions in and out of context (D.P. Bellamy) 31. An update on the elusive fixed-point property (C.L. Hagopian) 32. Hyperspaces of continua (A. Ilanes) 33. Inverse limits and dynamical systems (W.T. Ingram) 34. Indecomposable continua (W. Lewis) 35. Open problems on dendroids (V. Martinez-de-la-Vega and J.M. Martinez-Montejano) 36. 1/2-Homogeneous continua (S.B. Nadler, Jr.) 37. Thirty open problems in the theory of homogeneous continua (J.R. Prajs)
• Part 4. Topological Algebra 38. Problems about the uniform structures of topological groups (A. Bouziad and J-P. Troallic) 39. On some special classes of continuous maps (M.M. Clementino and D. Hofmann) 40. Dense subgroups of compact groups (W.W. Comfort) 41. Selected topics from the structure theory of topological groups (D. Dikranjan and D. Shakhmatov) 42. Recent results and open questions relating Chu duality and Bohr compactifications of locally compact groups (J. Galindo, S. Hernandez and T-S. Wu) 43. Topological transformation groups: selected topics (M. Megrelishvili) 44. Forty-plus annotated questions about large topological groups (V. Pestov)
• Part 5. Dynamical Systems 45. Minimal flows (W.F. Basener, K. Parwani and T. Wiandt) 46. The dynamics of tiling spaces (A. Clark) 47. Open problems in complex dynamics and "complex" topology (R.L. Devaney) 48. The topology and dynamics of flows (M.C. Sullivan)
• Part 6. Computer Science 49. Computational topology (D. Blackmore and T.J. Peters)
• Part 7. Functional Analysis 50. Non-smooth analysis, optimisation theory and Banach space theory (J.M. Borwein and W.B. Moors) 51. Topological structures of ordinary differential equations (V.V. Filippov) 52. The interplay between compact spaces and the Banach spaces of their continuous functions (P. Koszmider) 53. Tightness and t-equivalence (O. Okunev) 54. Topological problems in nonlinear and functional analysis (B. Ricceri) 55. Twenty questions on metacompactness in function spaces (V.V. Tkachuk)
• Part 8. Dimension Theory 56. Open problems in infinite-dimensional topology (T. Banakh, R. Cauty and M. Zarichnyi) 57. Classical dimension theory (V.A. Chatyrko) 58. Questions on weakly infinite-dimensional spaces (V.V. Fedorchuk) 59. Some problems in the dimension theory of compacta (B.A. Pasynkov)
• Part 9. Invited Papers 60. Problems from the Lviv topological seminar (T. Banakh, B. Bokalo, I. Guran, T. Radul and M. Zarichnyi) 61. Problems from the Bizerte-Sfax-Tunis Seminar (O. Echi, H. Marzougui and E. Salhi) 62. Cantor set problems (D.J. Garity and D. Repovs) 63. Problems from the Galway Topology Colloquium (C. Good, A. Marsh, A. McCluskey and B. McMaster) 64. The lattice of quasi-uniformities (E.P. de Jager amd H-P.A. Kunzi) 65. Topology in North Bay: some problems in continuum theory, dimension theory and selections (A. Karasev, M. Tuncali and V. Valov) 66. Moscow questions on topological algebra (K.L. Kozlov, E.A. Reznichenko and O.V. Sipacheva) 67. Some problems from George Mason University (J. Kulesza, R. Levy and M. Matveev) 68. Some problems on generalized metrizable spaces (S. Lin) 69. Problems from the Madrid Department of Geometry and Topology (J.M.R. Sanjurjo) 70. Cardinal sequences and universal spaces (L. Soukup) List of contributors Index.
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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.
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• A Crisis in the Experimental Method Archetype-- Orbit Organization in R2 x S1-- Braids as Indicators of Phase-Space Dynamics-- Braids and the Poincare Section-- Reconstruction of Phase-Space Dynamics
• Basic Course-- Reconstruction of Phase-Space Dynamics
• Advanced Course.
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This book presents the development and application of some topological methods in the analysis of data coming from 3D dynamical systems (or related objects). The aim is to emphasize the scope and limitations of the methods, what they provide and what they do not provide. Braid theory, the topology of surface homeomorphisms, data analysis and the reconstruction of phase-space dynamics are thoroughly addressed.
(source: Nielsen Book Data)

• PART I: A GEOMETRIC INTRODUCTION TO TOPOLOGY
• PART II: COVERING SPACES, CW COMPLEXES AND HOMOLOGY.
• (source: Nielsen Book Data)

This new-in-paperback introduction to topology emphasizes a geometric approach with a focus on surfaces. A primary feature is a large collection of exercises and projects, which fosters a teaching style that encourages the student to be an active class participant. A wide range of material at different levels supports flexible use of the book for a variety of students. Part I is appropriate for a one-semester or two-quarter course, and Part II (which is problem based) allows the book to be used for a year-long course which supports a variety of syllabuses. The over 750 exercises range from simple checks of omitted details in arguments, to reinforce the material and increase student involvement, to the development of substantial theorems that have been broken into many steps. The style encourages an active student role. Solutions to selected exercises are included as an appendix, with solutions to all exercises available to the instructor on a companion website.
(source: Nielsen Book Data)

• PART I: A GEOMETRIC INTRODUCTION TO TOPOLOGY
• PART II: COVERING SPACES, CW COMPLEXES AND HOMOLOGY.
• (source: Nielsen Book Data)

This new-in-paperback introduction to topology emphasizes a geometric approach with a focus on surfaces. A primary feature is a large collection of exercises and projects, which fosters a teaching style that encourages the student to be an active class participant. A wide range of material at different levels supports flexible use of the book for a variety of students. Part I is appropriate for a one-semester or two-quarter course, and Part II (which is problem based) allows the book to be used for a year-long course which supports a variety of syllabuses. The over 750 exercises range from simple checks of omitted details in arguments, to reinforce the material and increase student involvement, to the development of substantial theorems that have been broken into many steps. The style encourages an active student role. Solutions to selected exercises are included as an appendix, with solutions to all exercises available to the instructor on a companion website.
(source: Nielsen Book Data)

• Preface.- Introduction.- Set Theory.- Metric Spaces.- Set Theoretic Topology.- Systems of Continuous Functions.- Basic Algebraic Topology.- The Classical Mittag-Leffler Theorem Derived from Bourbaki?s.- Failure of the Heine-Borel Theorem in Infinite-Dimensional Spaces.- The Arzela-Ascoli Theorem.- References.- List of Symbols.- Index.
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If mathematics is a language, then taking a topology course at the undergraduate level is cramming vocabulary and memorizing irregular verbs: a necessary, but not always exciting exercise one has to go through before one can read great works of literature in the original language. The present book grew out of notes for an introductory topology course at the University of Alberta. It provides a concise introduction to set theoretic topology (and to a tiny little bit of algebraic topology). It is accessible to undergraduates from the second year on, but even beginning graduate students can benefit from some parts. Great care has been devoted to the selection of examples that are not self-serving, but already accessible for students who have a background in calculus and elementary algebra, but not necessarily in real or complex analysis.In some points, the book treats its material differently than other texts on the subject: Baire's theorem is derived from Bourbaki's Mittag-Leffler theorem; nets are used extensively, in particular for an intuitive proof of Tychonoff's theorem; and, a short and elegant, but little known proof for the Stone-Weierstrass theorem is given.
(source: Nielsen Book Data)