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From Geometry to Topologyby H Graham Flegg
Synopses & ReviewsPublisher Comments:An essential element of advanced studies in mathematics, topology tends to receive a highly formal and abstract treatment, discouraging students from grasping even the simpler ideas or getting any real "feel" for the subject. This volume, on the other hand, offers students a bridge from the familiar concepts of geometry to the formalized study of topology. It begins by exploring simple transformations of familiar figures in ordinary Euclidean space and develops the idea of congruence classes. By gradually expanding the number of "permitted" transformations, these classes increase and their relationships to topological properties develop in an intuitive manner. Imaginative introductions to selected topological subjects complete the intuitive approach, and students then advance to a more conventional presentation. An invaluable initiation into the formal study of topology for prospective and firstyear mathematics students.
Synopsis:Introductory text for firstyear math students uses intuitive approach, bridges the gap from familiar concepts of geometry to topology. Exercises and Problems. Includes 101 blackandwhite illustrations. 1974 edition. Synopsis:This excellent introduction to topology eases firstyear 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 blackandwhite illustrations. 1974 edition. Synopsis:This introduction to topology eases readers into the subject by building a bridge from the familiar concepts of geometry to the formalized study of topology. Focuses on congruence classes defined by transformations in real Euclidean space, continuity, sets, functions, metric spaces, and topological spaces, and more. Exercises and Problems. Includes 101 blackandwhite illustrations. 1974 edition. Table of Contents Author's Preface
Acknowledgements 1 Congruence Classes What geometry is about Congruence "The rigid transformations: translation, reflection, rotation" Invariant properties Congruence as an equivalence relation Congruence classes as the concern of Euclidean geometry 2 NonEuclidean Geometries Orientation as a property Orientation geometry divides congruence classes Magnification (and contraction) combine congruence classes Invariants of similarity geometry Affine and projective transformations and invariants Continuing process of combining equivalence classes 3 From Geometry to Topology Elastic deformations Intuitive idea of preservation of neighbourhoods Topological equivalence classes Derivation of 'topology' Close connection with study of continuity 4 Surfaces Surface of sphere "Properties of regions, paths and curves on a sphere" Similar considerations for torus and nfold torus Separation of surface by curves Genus as a topological property Closed and open surfaces Twosided and onesided surfaces Special surfaces: Moebius band and Klein bottle Intuitive idea of orientability Important properties remain under oneone bicontinuous transformations 5 Connectivity Further topological properties of surfaces Connected and disconnected surfaces Connectivity Contraction of simple closed curves to a point Homotopy classes Relation between homotopy classes and connectivity Cuts reducing surfaces to a disc Rank of open and closed surfaces Rank of connectivity 6 Euler Characteristic Maps "Interrelation between vertices, arcs and regions" Euler characteristic as a topological property Relation with genus Flow on a surface "Singular points: sinks, sources, vortices, etc." Index of a singular point Singular points and Euler characteristic 7 Networks Netowrks Odd and even vertices Planar and nonplanar networks Paths through networks Connected and disconnected networks Trees and cotrees Specifying a network: cutsets and tiesets Traversing a network The Koenigsberg Bridge problem and extensions 8 The Colouring of Maps Colouring maps Chromatic number Regular maps Six colour theorem General relation to Euler characteristic Five colour theorem for maps on a sphere 9 The Jordan Curve Theorem Separating properties of simple closed curves Difficulty of general proof Definition of inside and outside Polygonal paths in a plane Proof of Jordan curve theorem for polygonal paths 10 Fixed Point Theorems Rotating a disc: fixed point at centre Contrast with annulus Continuous transformation of disc to itself Fixed point principle Simple onedimensional case Proof based on labelling line segments Twodimensional case with triangles Threedimensional case with tetrahedra 11 Plane Diagrams Definition of manifold Constructions of manifolds from rectangle "Plane diagram represenations of sphere, torus, Moebius band, etc. " The real projective plane Euler characteristic from plane diagrams Seven colour theorem on a torus Symbolic representation of surfaces Indication of open and closed surfaces Orientability 12 The Standard Model Removal of disc from a sphere Addition of handles Standard model of twosided surfaces Addition of crosscaps General standard model Rank Relation to Euler characteristic Decomposition of surfaces "General classification as open or closed, twosided or onesided" Homeomorphic classes 13 Continuity Preservation of neighbourhood Distrance Continuous an discontinuous curves Formal definition of distance Triangle inequality Distance in ndimensional Euclidean space Formal definition of neighbourhood ed definition of continuity at a point Definition of continuous transformation 14 The Language of Sets Sets and subsets defined Set equality Null set Power set Union and Intersection Complement Laws of set theory Venn diagrams Index sets Infinite Intervals Cartesian product ndimensional Euclidean space 15 Functions Definition of function Domain and codomain Image and image set "Injection, bijection, surjection" Examples of functions as transformations Complex functions Inversion Point at infinity Bilinear functions Inverse functions Identity function "Open, closed, and halfopen subsets of R " Tearing by discontinuous functions 16 Metric Spaces Distance in Rn Definition of metric Neighbourhoods Continuity in terms of neighbourhoods Complete system of neighbourhoods Requirement for proof of noncontinuity Functional relationships between d and e Limitations of metric 17 Topological Spaces Concept of open set Definition of a topology on a set Topological space Examples of topological spaces Open and closed sets Redefining neighbourhood Metrizable topological spaces Closure "Interior, exterior, boundary" Continuity in terms of open sets Homeomorphic topological spaces Connected and disconnected spaces Covering Compactness Completeness: not a topological property Completeness of the real numbers "Topology, the starting point of real analysis" Historical Note Exercises and Problems Bibliography Index What Our Readers Are SayingBe the first to add 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