THE

BLACK ARTS OF TOPOLOGY with HOMOTOPY

by Scott W. Williams

Modern General Toplogy with Dynamics and Homotopy is a working title of a textbook, known informally as THE BLACK ARTS OF TOPOLOGY with HOMOTOPY, by Scott W. WILLIAMS to be published by John Wiley & Sons. Projected date of appearance is the Fall of 2000.

e-CONTACT Scott W. Williams

snail mail to:

• Scott W. Williams
• Professor of Mathematics
• State University of New York at Buffalo
• Buffalo NY 14214 USA

Even the table of contents is still subject to changes. In a subsequent version of this document, I will include important results from each section.

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TABLE OF CONTENTS

PART 0. Introduction

• 1. Preface
• 2. Conventions.
• 3. A course with Homotopy.
• 3. A courses with Dynamics.
• 4. A General Topology course.
• 5. A graduate course without Set Theory

PART I. Fundamentals and Set Theory

• 1. Some Conventions
• 2. Relations
• 3. ZFC - The Axioms of Set Theory
• 4. Partial orders
• 5. Cardinals and ordinals
• 6. Embeddings and order-isomorphisms
• 7. Applications of choice
• 8. Some combinatorics
• 9. CH and MA

PART II. General Topology

• 1. R and order generalizations
• 2. R^2 and distance generalizations
• 3. Topologies
• 4. Convergence
• 5. Axioms of countability

PART III. Functions

• 1. continuity and homeomorphisms
• 2. quotients and actions
• 3. products spaces
• 4. more new spaces from old
• 5. connected sets

PART IV. Separation Axioms

• 1. Kolmogoroff and Riesz
• 2. Hausdorff spaces
• 3. regular spaces and the absolute
• 4. Tychonov and uniform spaces
• 5. normal spaces
• 6. non-normal spaces
• 7. Very strong separation axioms

PART V. Covering axioms

• 1. paracompact spaces
• 2. partitions of unity and barycentric refinements
• 3. Equivalences of paracompact and the Stone Coincidence
• 4. strong paracompactness properties
• 5. weak paracompactness properties

PART VI. Compactness

• 1. compact spaces
• 2. compactness in other classes
• 3. perfect and irreducible maps
• 4. locally compact, Cech-complete, and sigma-compact spaces
• 5. Baire spaces and the Baire Category Theorem
• 6. pseudocompact and countably compact spaces
• 7. Compactifications
• 8. ßX
• 9. ßX\X
• 10. products with compact factors

PART VII. Metric spaces revisited

• 1. some metrization theorems
• 2. complete metric spaces and completions
• 3. completely metrizable spaces
• 4. totally bounded metric spaces
• 5. compact metric spaces
• 6. the rationals, the irrationals, and the Cantor set
• 7. more subspaces of the line

PART VIII. Continua (under construction)

• 1. Peano spaces
• 2. the line and the circle
• 3. simple closed curves
• 4. manifolds
• 5. surfaces: compact connected 2-manifolds.

PART IX. General dynamics

• 1. basic constructions
• 2. fixed points
• 3. periodic and almost periodic points
• 4. recurrence
• 5. Multiple Birkhoff recurrence
• 6. applications to algebra and combinatorics
• 7. applications from continua theory

PART X. Homotopy (under construction)

• 1. Basic notions
• 2. An algebra of paths
• 3. first computations
• 4. liftings
• 5. breakfast, lunch, and the world
• 6. Seifert-vanKampfen
• 7. surfaces
• 8. lots of knots
• 9. higher homotopy groups

APPENDICES (under construction)

• 1. More set theory
• 2 Applications of elementary submodels to topology. (tentative)
• 3. Function spaces and hyperspaces
• 4. Dimension
• 5. Groups, groupoids, and semi-groups
• 6. Hints to some problems
• References
• Index

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