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