Author's Preface |
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Acknowledgements |
1 |
Congruence Classes |
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What geometry is about |
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Congruence |
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"The rigid transformations: translation, reflection, rotation" |
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Invariant properties |
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Congruence as an equivalence relation |
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Congruence classes as the concern of Euclidean geometry |
2 |
Non-Euclidean Geometries |
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Orientation as a property |
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Orientation geometry divides congruence classes |
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Magnification (and contraction) combine congruence classes |
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Invariants of similarity geometry |
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Affine and projective transformations and invariants |
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Continuing process of combining equivalence classes |
3 |
From Geometry to Topology |
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Elastic deformations |
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Intuitive idea of preservation of neighbourhoods |
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Topological equivalence classes |
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Derivation of 'topology' |
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Close connection with study of continuity |
4 |
Surfaces |
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Surface of sphere |
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"Properties of regions, paths and curves on a sphere" |
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Similar considerations for torus and n-fold torus |
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Separation of surface by curves |
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Genus as a topological property |
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Closed and open surfaces |
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Two-sided and one-sided surfaces |
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Special surfaces: Moebius band and Klein bottle |
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Intuitive idea of orientability |
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Important properties remain under one-one bicontinuous transformations |
5 |
Connectivity |
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Further topological properties of surfaces |
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Connected and disconnected surfaces |
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Connectivity |
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Contraction of simple closed curves to a point |
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Homotopy classes |
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Relation between homotopy classes and connectivity |
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Cuts reducing surfaces to a disc |
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Rank of open and closed surfaces |
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Rank of connectivity |
6 |
Euler Characteristic |
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Maps |
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"Interrelation between vertices, arcs and regions" |
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Euler characteristic as a topological property |
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Relation with genus |
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Flow on a surface |
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"Singular points: sinks, sources, vortices, etc." |
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Index of a singular point |
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Singular points and Euler characteristic |
7 |
Networks |
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Netowrks |
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Odd and even vertices |
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Planar and non-planar networks |
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Paths through networks |
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Connected and disconnected networks |
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Trees and co-trees |
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Specifying a network: cutsets and tiesets |
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Traversing a network |
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The Koenigsberg Bridge problem and extensions |
8 |
The Colouring of Maps |
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Colouring maps |
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Chromatic number |
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Regular maps |
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Six colour theorem |
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General relation to Euler characteristic |
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Five colour theorem for maps on a sphere |
9 |
The Jordan Curve Theorem |
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Separating properties of simple closed curves |
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Difficulty of general proof |
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Definition of inside and outside |
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Polygonal paths in a plane |
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Proof of Jordan curve theorem for polygonal paths |
10 |
Fixed Point Theorems |
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Rotating a disc: fixed point at centre |
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Contrast with annulus |
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Continuous transformation of disc to itself |
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Fixed point principle |
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Simple one-dimensional case |
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Proof based on labelling line segments |
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Two-dimensional case with triangles |
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Three-dimensional case with tetrahedra |
11 |
Plane Diagrams |
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Definition of manifold |
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Constructions of manifolds from rectangle |
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"Plane diagram represenations of sphere, torus, Moebius band, etc. " |
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The real projective plane |
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Euler characteristic from plane diagrams |
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Seven colour theorem on a torus |
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Symbolic representation of surfaces |
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Indication of open and closed surfaces |
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Orientability |
12 |
The Standard Model |
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Removal of disc from a sphere |
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Addition of handles |
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Standard model of two-sided surfaces |
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Addition of cross-caps |
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General standard model |
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Rank |
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Relation to Euler characteristic |
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Decomposition of surfaces |
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"General classification as open or closed, two-sided or one-sided" |
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Homeomorphic classes |
13 |
Continuity |
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Preservation of neighbourhood |
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Distrance |
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Continuous an discontinuous curves |
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Formal definition of distance |
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Triangle in-equality |
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Distance in n-dimensional Euclidean space |
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Formal definition of neighbourhood |
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e-d definition of continuity at a point |
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Definition of continuous transformation |
14 |
The Language of Sets |
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Sets and subsets defined |
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Set equality |
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Null set |
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Power set |
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Union and Intersection |
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Complement |
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Laws of set theory |
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Venn diagrams |
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Index sets |
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Infinite |
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Intervals |
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Cartesian product |
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n-dimensional Euclidean space |
15 |
Functions |
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Definition of function |
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Domain and codomain |
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Image and image set |
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"Injection, bijection, surjection" |
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Examples of functions as transformations |
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Complex functions |
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Inversion |
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Point at infinity |
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Bilinear functions |
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Inverse functions |
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Identity function |
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"Open, closed, and half-open subsets of R " |
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Tearing by discontinuous functions |
16 |
Metric Spaces |
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Distance in Rn |
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Definition of metric |
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Neighbourhoods |
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Continuity in terms of neighbourhoods |
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Complete system of neighbourhoods |
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Requirement for proof of non-continuity |
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Functional relationships between d and e |
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Limitations of metric |
17 |
Topological Spaces |
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Concept of open set |
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Definition of a topology on a set |
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Topological space |
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Examples of topological spaces |
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Open and closed sets |
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Redefining neighbourhood |
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Metrizable topological spaces |
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Closure |
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"Interior, exterior, boundary" |
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Continuity in terms of open sets |
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Homeomorphic topological spaces |
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Connected and disconnected spaces |
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Covering |
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Compactness |
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Completeness: not a topological property |
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Completeness of the real numbers |
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"Topology, the starting point of real analysis" |
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Historical Note |
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Exercises and Problems |
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Bibliography |
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Index |
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