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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