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Publisher:

Dover Publications

Publication Date:

2001

Number of Pages:

208

Format:

Paperback

Price:

13.95

ISBN:

9780486419619

Category:

Textbook

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

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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 | ||||||||

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