From Geometry to Topology

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 Non-Euclidean 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 n-fold torus Separation of surface by curves Genus as a topological property Closed and open surfaces Two-sided and one-sided surfaces Special surfaces: Moebius band and Klein bottle Intuitive idea of orientability Important properties remain under one-one 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 non-planar networks Paths through networks Connected and disconnected networks Trees and co-trees 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 one-dimensional case Proof based on labelling line segments Two-dimensional case with triangles Three-dimensional 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 two-sided surfaces Addition of cross-caps General standard model Rank Relation to Euler characteristic Decomposition of surfaces "General classification as open or closed, two-sided or one-sided" Homeomorphic classes 13 Continuity Preservation of neighbourhood Distrance Continuous an discontinuous curves Formal definition of distance Triangle in-equality Distance in n-dimensional Euclidean space Formal definition of neighbourhood e-d 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 n-dimensional 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 half-open 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 non-continuity 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