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From Geometry to Topology

H. Graham Flegg
Dover Publications
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Author's Preface
1 Congruence Classes
    What geometry is about
    "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
    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
    "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
    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
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
    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
    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
    Laws of set theory
    Venn diagrams
    Index sets
    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
    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
    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
    "Interior, exterior, boundary"
    Continuity in terms of open sets
    Homeomorphic topological spaces
    Connected and disconnected spaces
    Completeness: not a topological property
    Completeness of the real numbers
    "Topology, the starting point of real analysis"
  Historical Note
  Exercises and Problems