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 
