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.

Acknowledgments 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 Matter45

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

eigh borhoods 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

Weigh borhoods, Open Sets, and Topologies Again 97

Finite-Point Spaces and Geometrical Interpretability 99

A Ceo metrically Z/n interpretable 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

jp-e lose d and Jp-open Sets115

JP-open Sets and C pen Sets, Jp-e lose d Sets and Closed Sets 117

Zeno's Combs 120

Closed Sets, C pen Sets, and Complements 123

Interiors, Boundary Points, and Boundaries 127

Formal Properties of Boundary Points 136

Con nee ted Spaces 140

Chains and Con nee ted ness 143

Directedness and Connectedness 148

Exercises 150

4.Separation Properties,Convergence, and Extensions 152

Separation Properties 152

Convergence and Unpleasantness 155

Sequences and Converge nee 160

S xtensions 163

The To polo gist's Sine Curve 165

PhysicalInterlude:Thomson'sLamp 168

Exercises 172

5.Properties of Functions 174

Continuity:an Overview 174

The Intuitive Explication of Continuity and Its Shortcomings 175

The Standard De inition 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;/i neal 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

EudoxanandAnthyphairetic 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:ARe mark 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

eigh borhoods 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 EachOther? 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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