Open Problems in Topology II.

By: Pearl, ElliottContributor(s): Pearl, Elliott MPublisher: Oxford : Elsevier Science & Technology, 2007Copyright date: ©2007Description: 1 online resource (777 pages)Content type: text Media type: computer Carrier type: online resourceISBN: 9780080475295Subject(s): TopologyGenre/Form: Electronic books. Additional physical formats: Print version:: Open Problems in Topology IIDDC classification: 514 LOC classification: QA611.O562 2007Online resources: Click to View
Contents:
Front Cover -- Open Problems in Topology II -- Copyright Page -- Preface -- Table of Contents -- Part 1. General Topology -- Chapter 1. Selected ordered space problems -- 1. Introduction -- 2. A few of our favorite things -- References -- Chapter 2. Problems on star-covering properties -- Introduction -- Compactness-type properties -- Lindelöf-type properties and cardinal functions -- Paracompactness-type properties -- Property (a) -- References -- Chapter 3. Function space topologies -- 1. Splitting and admissible topologies -- 2. The greatest splitting, compact-open, and Isbell topologies -- References -- Chapter 4. Spaces and mappings: special networks -- Introduction -- Mappings -- k-networks and weak bases -- Spaces determined or dominated by certain covers -- Products of k-spaces having certain k-networks -- References -- Chapter 5. Extension problems of real-valued continuous functions -- 1. Introduction -- 2. C-embedding versus C*-embedding -- 3. π-embedding and πZ-embeddings -- 4. Miscellaneous questions -- References -- Chapter 6. LΣ(k)-spaces -- References -- Chapter 7. Problems on (ir)resolvability -- Introduction. Resolvability hierarchy -- Resolvability of connected spaces -- Baire property and a problem of Katětov -- Groups and homogeneous spaces -- Compactness, products, miscellaneous questions -- Structure resolvability -- Literature -- References -- Chapter 8. Topological games and Ramsey theory -- Introduction -- 1. Basic concepts -- 2. The point-picking ganes -- 3. Modifications of the point-picking games -- 4. The set-picking games -- 5. Further restrictions on ONE -- 6. Banach-Mazur games -- 7. The length of games -- 8. Memory restrictions on the winning player -- 9. Ramsey theory -- References -- Chapter 9. Selection principles and special sets of reals -- 1. Introduction -- 2. The Scheepers Diagram problem.
3. Examples without special set-theoretic hypotheses -- 4. Examples from CH or MA -- 5. The δ-property -- 6. Preservation of properties -- 7. Modern types of covers -- 8. Splittability -- 9. Function spaces and local-global principles -- 10. Topological groups -- 11. Cardinal characteristics of the continuum -- 12. Additional problems and other special sets of reals -- References -- Part 2. Set-theoretic Topology -- Chapter 10. Introduction: Twenty problems in set-theoretic topology -- References -- Chapter 11. Thin-tall spaces and cardinal sequences -- 1. Introduction -- 2. Spaces of countable height -- 3. Spaces of height ω2 -- 6. A basic construction -- 7. Forcing an LCS space -- 8. Final remarks -- References -- Chapter 12. Sequential order -- References -- Chapter 13. On D-spaces -- Introduction -- Questions about D-spaces -- Stickyness -- General remarks -- References -- Chapter 14. The fourth head of βN -- 1. Trivial continuous maps -- 2. A partial result -- 3. Rigidity phenomena for quotients P(N)/I -- 4. Further results -- 5. Conclusion -- References -- Chapter 15. Are stratifiable spaces M1? -- 1. Equivalent questions -- 2. Related classes and partial results -- 3. Function spaces and a possible counterexample -- 4. Some final remarks -- References -- Chapter 16. Perfect compacta and basis problems in topology -- 1. Perfect compacta -- 2. Uncountable spaces -- 3. Approaches, axiomatics, further reading -- References -- Chapter 17. Selection problems for hyperspaces -- 1. Weak selections and orderability -- 2. Topological well-ordering and selections -- 3. Selections and disconnectedness-like properties -- References -- Chapter 18. Efimov's problem -- Introduction -- 1. Attacking the problem -- 2. Counterexamples -- 3. Is there still a problem? -- 4. Larger cardinals -- References.
Chapter 19. Completely separable MAD families -- 1. The main problem -- 2. Topological connection -- 3. MAD families in forcing extensions -- References -- Chapter 20. Good, splendid and Jakovlev -- 1. The size of good and splendid spaces -- 2. Connections to other problems -- References -- Chapter 21. Homogeneous compacta -- 0. Introduction -- 1. Rudin's problem -- 2. Van Douwen's problem -- 3. Arhangeľskiĭ's problem -- 4. Continuous images of homogeneous compacta -- 5. Remarks -- References -- Chapter 22. Compact spaces with hereditarily normal squares -- References -- Chapter 23. The metrization problem for Fréchet groups -- 1. Introduction -- 2. The role of gaps -- 3. Other convergence properties -- 4. In search of a test model -- References -- Chapter 24. Čech-Stone remainders of discrete spaces -- 1. Introduction -- 2. Some basics -- 3. Some consequences of Axiom Ω -- 4. Implications for ω1* -- 5. Some more open problems about Axiom Ω -- 6. Notes on Problem 2 -- References -- Chapter 25. First countable, countably compact, noncompact spaces -- 1. Consistent good examples for Problem 1 -- 2. Other consistent constructions for Problem 1 -- 3. Arbitrarily large first countable, locally compact, countably compact spaces -- 4. Towards negative answers -- References -- Chapter 26. Linearly Lindelöf problems -- Introduction -- Local compactness -- Sequentially linearly Lindelöf spaces -- Discretely Lindelöf spaces -- Estimating cardinality -- The Hušek number -- A consistent realcompact example -- A consistent first countable example -- Lindelöf problems -- Acknowledgments -- References -- Chapter 27. Small Dowker spaces -- References -- Chapter 28. Reflection of topological properties to Aleph1 -- References -- Chapter 29. The Scarborough-Stone problem -- 1. Introduction -- 2. How to construct a product that is not countably compact.
3. The Ostaszewski construction -- 4. Franklin-Rajagopalan spaces -- 5. Martin's axiom and proper forcing -- 6. Concluding comments -- References -- Part 3. Continuum Theory -- Chapter 30. Questions in and out of context -- References -- Chapter 31. An update on the elusive fixed-point property -- References -- Chapter 32. Hyperspaces of continua -- Introduction -- Whitney and diameter maps -- Cones, products and hyperspaces -- Means -- Fixed point property -- Mappings between hyperspaces -- Unicoherence of F2(S1) -- Locating cells in hyperspaces -- Unique hyperspaces -- 1. Miscellaneous problems -- References -- Chapter 33 Inverse limits and dynamical systems -- 1. Introduction -- 2. Characterization of chainability -- 3. Plane embedding -- 4. Inverse limits on [0, 1] -- 5. The property of Kelley -- 6. Inverse limits with upper semi-continuous bonding functions -- 7. Applications of inverse limits in Economics -- References -- Chapter 34. Indecomposable continua -- Hereditary equivalence -- Homogeneity -- Epsilon-premaps -- Fixed points -- Maps of products -- Homeomorphism groups -- Q-like continua -- Hyperspaces -- Dimensions greater than one -- Non-metric continua -- Conclusion -- References -- Chapter 35. Open problems on dendroids -- 1. Introduction -- 2. The problem -- 3. Mappings on dendrites -- 4. Maps onto dendroids -- 5. Contractibility -- 6. Hyperspaces -- 7. Property of Kelley -- 8. Retractions -- 9. Means -- 10. Selections -- 11. Smooth dendroids -- 12. Planability -- 13. Shore sets -- References -- Chapter 36. 1/2-Homogeneous continua -- 1. Introduction -- 2. Notation and terminology -- 3. 1/2-Homogeneous continua with cut points -- 4. 1/2-Homogeneous cones -- 5. 1/2-Homogeneous hyperspaces -- References -- Chapter 37. Thirty open problems in the theory of homogeneous continua -- 1. Preliminaries -- 2. Fourteen miscellaneous problems.
3. Filament sets: definitions and basic properties -- 4. Filament sets: sixteen questions -- References -- Part 4. Topological Algebra -- Chapter 38. Problems about the uniform structures of topological groups -- 1. Introduction -- 2. The two versions of Itzkowitz's problem -- 3. Some remarks about [FSIN] and [SIN] -- 4. The class [ASIN] -- 5. A few other questions -- 6. A representative case -- References -- Chapter 39. On some special classes of continuous maps -- 1. Special morphisms of Top -- 2. Corresponding morphisms in related categories -- References -- Chapter 40. Dense subgroups of compact groups -- 1. Introduction -- 2. Groups with topologies of pre-assigned type -- 3. Topologies induced by groups of characters -- 4. Extremal phenomena -- 5. Miscellaneous questions -- References -- Chapter 41. Selected topics from the structure theory of topological groups -- 1. Dimension theory of topological groups -- 2. Pseudocompact and countably compact group topologies on Abelian groups -- 3. Properties determined by convergent sequences -- 4. Categorically compact groups -- 5. The Markov-Zariski topology -- 6. Bohr topologies of Abelian groups -- 7. Miscellanea -- References -- Chapter 42. Recent results and open questions relating Chu duality and Bohr compactifications of locally compact groups -- 1. Introduction -- 2. Basic definitions -- 3. Abelian groups -- 4. Nonabelian groups -- 5. How is G placed in bG? Interpolation sets -- Acknowledgement -- References -- Chapter 43. Topological transformation groups: selected topics -- 1. Introduction -- 2. Equivariant compactifications -- 3. Equivariant normality -- 4. Universal actions -- 5. Free topological G-groups -- 6. Banach representations of groups -- 7. Dynamical versions of Eberlein and Radon-Nikodým compacta -- Acknowledgment -- References.
Chapter 44. Forty-plus annotated questions about large topological groups.
Summary: This volume is a collection of surveys of research problems in topology and its applications. The topics covered include general topology, set-theoretic topology, continuum theory, topological algebra, dynamical systems, computational topology and functional analysis. * New surveys of research problems in topology * New perspectives on classic problems * Representative surveys of research groups from all around the world.
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Front Cover -- Open Problems in Topology II -- Copyright Page -- Preface -- Table of Contents -- Part 1. General Topology -- Chapter 1. Selected ordered space problems -- 1. Introduction -- 2. A few of our favorite things -- References -- Chapter 2. Problems on star-covering properties -- Introduction -- Compactness-type properties -- Lindelöf-type properties and cardinal functions -- Paracompactness-type properties -- Property (a) -- References -- Chapter 3. Function space topologies -- 1. Splitting and admissible topologies -- 2. The greatest splitting, compact-open, and Isbell topologies -- References -- Chapter 4. Spaces and mappings: special networks -- Introduction -- Mappings -- k-networks and weak bases -- Spaces determined or dominated by certain covers -- Products of k-spaces having certain k-networks -- References -- Chapter 5. Extension problems of real-valued continuous functions -- 1. Introduction -- 2. C-embedding versus C*-embedding -- 3. π-embedding and πZ-embeddings -- 4. Miscellaneous questions -- References -- Chapter 6. LΣ(k)-spaces -- References -- Chapter 7. Problems on (ir)resolvability -- Introduction. Resolvability hierarchy -- Resolvability of connected spaces -- Baire property and a problem of Katětov -- Groups and homogeneous spaces -- Compactness, products, miscellaneous questions -- Structure resolvability -- Literature -- References -- Chapter 8. Topological games and Ramsey theory -- Introduction -- 1. Basic concepts -- 2. The point-picking ganes -- 3. Modifications of the point-picking games -- 4. The set-picking games -- 5. Further restrictions on ONE -- 6. Banach-Mazur games -- 7. The length of games -- 8. Memory restrictions on the winning player -- 9. Ramsey theory -- References -- Chapter 9. Selection principles and special sets of reals -- 1. Introduction -- 2. The Scheepers Diagram problem.

3. Examples without special set-theoretic hypotheses -- 4. Examples from CH or MA -- 5. The δ-property -- 6. Preservation of properties -- 7. Modern types of covers -- 8. Splittability -- 9. Function spaces and local-global principles -- 10. Topological groups -- 11. Cardinal characteristics of the continuum -- 12. Additional problems and other special sets of reals -- References -- Part 2. Set-theoretic Topology -- Chapter 10. Introduction: Twenty problems in set-theoretic topology -- References -- Chapter 11. Thin-tall spaces and cardinal sequences -- 1. Introduction -- 2. Spaces of countable height -- 3. Spaces of height ω2 -- 6. A basic construction -- 7. Forcing an LCS space -- 8. Final remarks -- References -- Chapter 12. Sequential order -- References -- Chapter 13. On D-spaces -- Introduction -- Questions about D-spaces -- Stickyness -- General remarks -- References -- Chapter 14. The fourth head of βN -- 1. Trivial continuous maps -- 2. A partial result -- 3. Rigidity phenomena for quotients P(N)/I -- 4. Further results -- 5. Conclusion -- References -- Chapter 15. Are stratifiable spaces M1? -- 1. Equivalent questions -- 2. Related classes and partial results -- 3. Function spaces and a possible counterexample -- 4. Some final remarks -- References -- Chapter 16. Perfect compacta and basis problems in topology -- 1. Perfect compacta -- 2. Uncountable spaces -- 3. Approaches, axiomatics, further reading -- References -- Chapter 17. Selection problems for hyperspaces -- 1. Weak selections and orderability -- 2. Topological well-ordering and selections -- 3. Selections and disconnectedness-like properties -- References -- Chapter 18. Efimov's problem -- Introduction -- 1. Attacking the problem -- 2. Counterexamples -- 3. Is there still a problem? -- 4. Larger cardinals -- References.

Chapter 19. Completely separable MAD families -- 1. The main problem -- 2. Topological connection -- 3. MAD families in forcing extensions -- References -- Chapter 20. Good, splendid and Jakovlev -- 1. The size of good and splendid spaces -- 2. Connections to other problems -- References -- Chapter 21. Homogeneous compacta -- 0. Introduction -- 1. Rudin's problem -- 2. Van Douwen's problem -- 3. Arhangeľskiĭ's problem -- 4. Continuous images of homogeneous compacta -- 5. Remarks -- References -- Chapter 22. Compact spaces with hereditarily normal squares -- References -- Chapter 23. The metrization problem for Fréchet groups -- 1. Introduction -- 2. The role of gaps -- 3. Other convergence properties -- 4. In search of a test model -- References -- Chapter 24. Čech-Stone remainders of discrete spaces -- 1. Introduction -- 2. Some basics -- 3. Some consequences of Axiom Ω -- 4. Implications for ω1* -- 5. Some more open problems about Axiom Ω -- 6. Notes on Problem 2 -- References -- Chapter 25. First countable, countably compact, noncompact spaces -- 1. Consistent good examples for Problem 1 -- 2. Other consistent constructions for Problem 1 -- 3. Arbitrarily large first countable, locally compact, countably compact spaces -- 4. Towards negative answers -- References -- Chapter 26. Linearly Lindelöf problems -- Introduction -- Local compactness -- Sequentially linearly Lindelöf spaces -- Discretely Lindelöf spaces -- Estimating cardinality -- The Hušek number -- A consistent realcompact example -- A consistent first countable example -- Lindelöf problems -- Acknowledgments -- References -- Chapter 27. Small Dowker spaces -- References -- Chapter 28. Reflection of topological properties to Aleph1 -- References -- Chapter 29. The Scarborough-Stone problem -- 1. Introduction -- 2. How to construct a product that is not countably compact.

3. The Ostaszewski construction -- 4. Franklin-Rajagopalan spaces -- 5. Martin's axiom and proper forcing -- 6. Concluding comments -- References -- Part 3. Continuum Theory -- Chapter 30. Questions in and out of context -- References -- Chapter 31. An update on the elusive fixed-point property -- References -- Chapter 32. Hyperspaces of continua -- Introduction -- Whitney and diameter maps -- Cones, products and hyperspaces -- Means -- Fixed point property -- Mappings between hyperspaces -- Unicoherence of F2(S1) -- Locating cells in hyperspaces -- Unique hyperspaces -- 1. Miscellaneous problems -- References -- Chapter 33 Inverse limits and dynamical systems -- 1. Introduction -- 2. Characterization of chainability -- 3. Plane embedding -- 4. Inverse limits on [0, 1] -- 5. The property of Kelley -- 6. Inverse limits with upper semi-continuous bonding functions -- 7. Applications of inverse limits in Economics -- References -- Chapter 34. Indecomposable continua -- Hereditary equivalence -- Homogeneity -- Epsilon-premaps -- Fixed points -- Maps of products -- Homeomorphism groups -- Q-like continua -- Hyperspaces -- Dimensions greater than one -- Non-metric continua -- Conclusion -- References -- Chapter 35. Open problems on dendroids -- 1. Introduction -- 2. The problem -- 3. Mappings on dendrites -- 4. Maps onto dendroids -- 5. Contractibility -- 6. Hyperspaces -- 7. Property of Kelley -- 8. Retractions -- 9. Means -- 10. Selections -- 11. Smooth dendroids -- 12. Planability -- 13. Shore sets -- References -- Chapter 36. 1/2-Homogeneous continua -- 1. Introduction -- 2. Notation and terminology -- 3. 1/2-Homogeneous continua with cut points -- 4. 1/2-Homogeneous cones -- 5. 1/2-Homogeneous hyperspaces -- References -- Chapter 37. Thirty open problems in the theory of homogeneous continua -- 1. Preliminaries -- 2. Fourteen miscellaneous problems.

3. Filament sets: definitions and basic properties -- 4. Filament sets: sixteen questions -- References -- Part 4. Topological Algebra -- Chapter 38. Problems about the uniform structures of topological groups -- 1. Introduction -- 2. The two versions of Itzkowitz's problem -- 3. Some remarks about [FSIN] and [SIN] -- 4. The class [ASIN] -- 5. A few other questions -- 6. A representative case -- References -- Chapter 39. On some special classes of continuous maps -- 1. Special morphisms of Top -- 2. Corresponding morphisms in related categories -- References -- Chapter 40. Dense subgroups of compact groups -- 1. Introduction -- 2. Groups with topologies of pre-assigned type -- 3. Topologies induced by groups of characters -- 4. Extremal phenomena -- 5. Miscellaneous questions -- References -- Chapter 41. Selected topics from the structure theory of topological groups -- 1. Dimension theory of topological groups -- 2. Pseudocompact and countably compact group topologies on Abelian groups -- 3. Properties determined by convergent sequences -- 4. Categorically compact groups -- 5. The Markov-Zariski topology -- 6. Bohr topologies of Abelian groups -- 7. Miscellanea -- References -- Chapter 42. Recent results and open questions relating Chu duality and Bohr compactifications of locally compact groups -- 1. Introduction -- 2. Basic definitions -- 3. Abelian groups -- 4. Nonabelian groups -- 5. How is G placed in bG? Interpolation sets -- Acknowledgement -- References -- Chapter 43. Topological transformation groups: selected topics -- 1. Introduction -- 2. Equivariant compactifications -- 3. Equivariant normality -- 4. Universal actions -- 5. Free topological G-groups -- 6. Banach representations of groups -- 7. Dynamical versions of Eberlein and Radon-Nikodým compacta -- Acknowledgment -- References.

Chapter 44. Forty-plus annotated questions about large topological groups.

This volume is a collection of surveys of research problems in topology and its applications. The topics covered include general topology, set-theoretic topology, continuum theory, topological algebra, dynamical systems, computational topology and functional analysis. * New surveys of research problems in topology * New perspectives on classic problems * Representative surveys of research groups from all around the world.

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