Tim Maudlin
Oxford University Press | 2014 | 374 páginas | rar - pdf | 1,94 Mb
link (password: matav)
Topology is the mathematical study of the most basic geometrical structure of a space. Mathematical physics uses topological spaces as the formal means for describing physical space and time. This book proposes a completely new mathematical structure for describing geometrical notions such as continuity, connectedness, boundaries of sets, and so on, in order to provide a better mathematical tool for understanding space-time. This is the initial volume in a two-volume set, the first of which develops the mathematical structure and the second of which applies it to classical and Relativistic physics.
The book begins with a brief historical review of the development of mathematics as it relates to geometry, and an overview of standard topology. The new theory, the Theory of Linear Structures, is presented and compared to standard topology. The Theory of Linear Structures replaces the foundational notion of standard topology, the open set, with the notion of a continuous line. Axioms for the Theory of Linear Structures are laid down, and definitions of other geometrical notions developed in those terms. Various novel geometrical properties, such as a space being intrinsically directed, are defined using these resources. Applications of the theory to discrete spaces (where the standard theory of open sets gets little purchase) are particularly noted. The mathematics is developed up through homotopy theory and compactness, along with ways to represent both affine (straight line) and metrical structure.
ContentsAcknowledgments x
Introduction 1
Metaphorical and Geometrical Spaces 6
A Light Dance on the Dust of the Ages 9
The Proliferation of Numbers 12
Descartes and Coordinate Geometry 14
John Wallis and the Number Line 16
Dedekind and the Construction of Irrational Numbers 20
Overview and Terminological Conventions 25
1. Topology and Its Shortcomings 28
Standard Topology 31
Closed Sets, Neighborhoods, Boundary Points, and Connected Spaces 33
The Hausdorff Property 36
Why Discrete Spaces Matter 45
The Relational Nature of Open Sets 47
The Bill of Indictment (So Far) 49
2. Linear Structures, Neighborhoods, Open Sets 54
Methodological Morals 54
The Essence of the Line 57
The (First) Theory of Linear Structures 59
Proto-Linear Structures 69
Discrete Spaces, Mr Bush’s Wild Line, the Woven Plane, and the Affine Plane 74
A Taxonomy of Linear Structures 79
Neighborhoods in a Linear Structure 81
Open Sets 85
Finite-Point Spaces 86
Return to Intuition 89
Directed Linear Structures 92
Linear Structures and Directed Linear Structures 96
Neighborhoods, Open Sets, and Topologies Again 97
Finite-Point Spaces and Geometrical Interpretability 99
A Geometrically Uninterpretable Topological Space 103
Segment-Spliced Linear Structures 104
Looking Ahead 107
Exercises 107
Appendix: Neighborhoods and Linear Structures 108
3. Closed Sets, Open Sets (Again), Connected Spaces 113
Closed Sets: Preliminary Observations 113
Open and Closed Intervals 114
IP-closed and IP-open Sets 115
IP-open Sets and Open Sets, IP-closed Sets and Closed Sets 117
Zeno’s Combs 120
Closed Sets, Open Sets, and Complements 123
Interiors, Boundary Points, and Boundaries 127
Formal Properties of Boundary Points 136
Connected Spaces 140
Chains and Connectedness 143
Directedness and Connectedness 148
Exercises 150
4. Separation Properties, Convergence, and Extensions 152
Separation Properties 152
Convergence and Unpleasantness 155
Sequences and Convergence 160
Extensions 163
The Topologist’s Sine Curve 165
Physical Interlude: Thomson’s Lamp 168
Exercises 172
5. Properties of Functions 174
Continuity: an Overview 174
The Intuitive Explication of Continuity and Its Shortcomings 175
The Standard Definition and Its Shortcomings 178
What the Standard Definition of “Continuity” Defines 183
The Essence of Continuity 186
Continuity at a Point and in a Direction 190
An Historical Interlude 192
Remarks on the Architecture of Definitions; Lineal Functions 194
Lines and Continuity in Standard Topology 199
Exercises 201
6. Subspaces and Substructures; Straightness and Differentiability 203
The Geometrical Structure of a Subspace: Desiderata 203
Subspaces in Standard Topology 205
Subspaces in the Theory of Linear Structures 206
Substructures 211
One Way Forward 218
Euclid’s Postulates and the Nature of Straightness 220
Convex Affine Spaces 227
Example: Some Conical Spaces 233
Tangents 235
Upper and Lower Tangents, Differentiability 244
Summation 253
Exercises 254
7. Metrical Structure 256
Approaches to Metrical Structure 256
Ratios Between What? 258
The Additive Properties of Straight Lines 260
Congruence and Comparability 262
Eudoxan and Anthyphairetic Ratios 274
The Compass 280
Metric Linear Structures and Metric Functions 285
Open Lines, Curved Lines, and Rectification 287
Continuity of the Metric 291
Exercises 294
Appendix: A Remark about Minimal Regular Metric Spaces 294
8. Product Spaces and Fiber Bundles 297
New Spaces from Old 297
Constructing Product Linear Structures 300
Examples of Product Linear Structures 303
Neighborhoods and Open Sets in Product Linear Structures 307
Fiber Bundles 309
Sections 313
Additional Structure 315
Exercises 318
9. Beyond Continua 320
How Can Continua and Non-Continua Approximate Each Other? 320
Continuous Functions 321
Homotopy 334
Compactness 339
Summary of Mathematical Results and Some Open Questions 345
Exercises 346
Axioms and Definitions 347
Bibliography 358
Index 361
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