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 1996.

EMAIL Scott W. Williams at:

- bonvibre@aol.com OR
- mthsww@ubvms.cc.buffalo.edu (before September 96) OR
- sww@acsu.buffalo.edu (after March 96)

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 AND INDEX**

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