| Edition |
1st ed |
| Descript |
ix, 363 p. : ill. ; 24 cm |
| Note |
Includes bibliographical references and index |
|
Machine generated contents note: Metaphorical and Geometrical Spaces -- A Light Dance on the Dust of the Ages -- The Proliferation of Numbers -- Descartes and Coordinate Geometry -- John Wallis and the Number Line -- Dedekind and the Construction of Irrational Numbers -- Overview and Terminological Conventions -- 1. Topology and Its Shortcomings -- Standard Topology -- Closed Sets, Neighborhoods, Boundary Points, and Connected Spaces -- The Hausdorff Property -- Why Discrete Spaces Matter -- The Relational Nature of Open Sets -- The Bill of Indictment (So Far) -- 2. Linear Structures, Neighborhoods, Open Sets -- Methodological Morals -- The Essence of the Line -- The (First) Theory of Linear Structures -- Proto-Linear Structures -- Discrete Spaces, Mr Bush's Wild Line, the Woven Plane, and the Affine Plane -- A Taxonomy of Linear Structures -- Neighborhoods in a Linear Structure -- Open Sets |
|
Finite-Point Spaces -- Return to Intuition -- Directed Linear Structures -- Linear Structures and Directed Linear Structures -- Neighborhoods, Open Sets, and Topologies Again -- Finite-Point Spaces and Geometrical Interpretability -- A Geometrically Uninterpretable Topological Space -- Segment-Spliced Linear Structures -- Looking Ahead -- Exercises -- Appendix: Neighborhoods and Linear Structures -- 3. Closed Sets, Open Sets (Again), Connected Spaces -- Closed Sets: Preliminary Observations -- Open and Closed Intervals -- JP-closed and JP-open Sets -- Jp-open Sets and Open Sets, JP-closed Sets and Closed Sets -- Zeno's Combs -- Closed Sets, Open Sets, and Complements -- Interiors, Boundary Points, and Boundaries -- Formal Properties of Boundary Points -- Connected Spaces -- Chains and Connectedness -- Directedness and Connectedness -- Exercises -- 4. Separation Properties, Convergence, and Extensions -- Separation Properties -- Convergence and Unpleasantness |
|
Sequences and Convergence -- Extensions -- The Topologist's Sine Curve -- Physical Interlude: Thomson's Lamp -- Exercises -- 5. Properties of Functions -- Continuity: an Overview -- The Intuitive Explication of Continuity and Its Shortcomings -- The Standard Definition and Its Shortcomings -- What the Standard Definition of "Continuity" Defines -- The Essence of Continuity -- Continuity at a Point and in a Direction -- An Historical Interlude -- Remarks on the Architecture of Definitions; Lineal Functions -- Lines and Continuity in Standard Topology -- Exercises -- 6. Subspaces and Substructures; Straightness and Differentiability -- The Geometrical Structure of a Subspace: Desiderata -- Subspaces in Standard Topology -- Subspaces in the Theory of Linear Structures -- Substructures -- One Way Forward -- Euclid's Postulates and the Nature of Straightness -- Convex Affine Spaces -- Example: Some Conical Spaces -- Tangents -- Upper and Lower Tangents, Differentiability -- Summation -- Exercises |
|
7. Metrical Structure -- Approaches to Metrical Structure -- Ratios Between What? -- The Additive Properties of Straight Lines -- Congruence and Comparability -- Eudoxan and Anthyphairetic Ratios -- The Compass -- Metric Linear Structures and Metric Functions -- Open Lines, Curved Lines, and Rectification -- Continuity of the Metric -- Exercises -- Appendix: A Remark about Minimal Regular Metric Spaces -- 8. Product Spaces and Fiber Bundles -- New Spaces from Old -- Constructing Product Linear Structures -- Examples of Product Linear Structures -- Neighborhoods and Open Sets in Product Linear Structures -- Fiber Bundles -- Sections -- Additional Structure -- Exercises -- 9. Beyond Continua -- How Can Continua and Non-Continua Approximate Each Other? -- Continuous Functions -- Homotopy -- Compactness -- Summary of Mathematical Results and Some Open Questions -- Exercises |
| Subject |
Raumstruktur. gnd
|
|
Topologischer Raum. gnd
|
|
Raum-Zeit. gnd
|
|
Geometry -- Philosophy
|
|
Algebras, Linear
|
|
Space and time -- Mathematics
|
|
Algebras, Linear. fast (OCoLC)fst00804946
|
|
Geometry -- Philosophy.
fast (OCoLC)fst00940874
|
|
Space and time -- Mathematics.
fast (OCoLC)fst01127629
|
|
