An elementary text that can be understood by anyone with a background in high school geometry, this text focuses on the problems inherent to coloring maps, homeomorphism, applications of Descartes' theorem, and topological polygons. Considerations of the topological classification of closed surfaces cover elementary operations, use of normal forms of polyhedra, more. Includes 108 figures. 1967 edition.
Author(s): Maurice Frechet, Ky Fan
Year: 2003
Language: English
Pages: 136
Cover......Page 1
Foreword......Page 5
Translator's Preface......Page 8
1 Qualitative Geometric Properties......Page 12
2 Coloring Geographical Maps......Page 13
3 The Problem of Neighboring Regions......Page 16
4 Topology, India-Rubber Geometry......Page 17
5 Homeomorphism......Page 18
6 Topology, Continuous Geometry......Page 22
7 Comparison of Elementary Geometry, Projective Geometry, & Topology......Page 23
8 Relative Topological Properties......Page 25
9 Set Topology & Combinatorial Topology......Page 28
10 The Development of Topology......Page 30
11 Descartes' Theorem......Page 32
12 An Application of Descartes' Theorem......Page 36
13 Characteristic of a Surface......Page 38
14 Unilateral Surfaces......Page 40
15 Orientability & Nonorientability......Page 42
16 Topological Polygons......Page 46
17 Construction of Closed Orientable Surfaces from Polygons by Identifying Their Sides......Page 47
18 Construction of closed Nonorientable Surfaces from Polygons by Identifying Their Sides......Page 51
19 Topological Definition of a Closed Surface......Page 56
20 The Principal Problem in the Topology of Surfaces......Page 60
21 Planar Polygonal Schema & Symbolic Representation of a Polyhedron......Page 61
22 Elementary Operations......Page 64
23 Use of Normal Forms of Polyhedra......Page 66
24 Reduction to Normal Form : I......Page 67
25 Reduction to Normal Form : II......Page 70
26 Characteristic & Orientability......Page 75
27 The Principal Theorem of the Topology of Closed Surfaces......Page 78
29 Genus & Connection Number of Closed Orientable Surfaces......Page 80
Bibliography......Page 84
TRANSLATOR'S NOTES......Page 86
Index......Page 131
