Set-Theoretic Topology (eBook)

George M. Reed (Herausgeber)

eBook Download: PDF

2014 | 1. Auflage
452 Seiten
Elsevier Science (Verlag)
978-1-4832-6392-2 (ISBN)

Infos

Set-Theoretic Topology deals with results concerning set theoretic topology and indicates directions for further investigations. Topics covered include normality and conditions in abstract spaces, compactifications, cardinal invariance, mapping theory, product spaces, and metrization. Comprised of 29 chapters, this volume begins with an example concerning the preservation of the Lindelöf property in product spaces, followed by a discussion on closed-completeness in spaces with a quasi-G? diagonal and with weak covering properties. The reader is then introduced to countably compact extensions of normal locally compact M-spaces; continuously semi-metrizable spaces; and closed discrete collections of singular cardinality. Subsequent chapters focus on open mapping theory; a selection-theoretic approach to certain extension theorems; semicompletable Moore spaces; and non-normal spaces. The book also considers complete mappings in base of countable order theory before concluding with an analysis of locally separable Moore spaces. This monograph should be of value to students, researchers, and specialists in the field of mathematics.

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Set-Theoretic Topology deals with results concerning set theoretic topology and indicates directions for further investigations. Topics covered include normality and conditions in abstract spaces, compactifications, cardinal invariance, mapping theory, product spaces, and metrization. Comprised of 29 chapters, this volume begins with an example concerning the preservation of the Lindelof property in product spaces, followed by a discussion on closed-completeness in spaces with a quasi-G? diagonal and with weak covering properties. The reader is then introduced to countably compact extensions of normal locally compact M-spaces; continuously semi-metrizable spaces; and closed discrete collections of singular cardinality. Subsequent chapters focus on open mapping theory; a selection-theoretic approach to certain extension theorems; semicompletable Moore spaces; and non-normal spaces. The book also considers complete mappings in base of countable order theory before concluding with an analysis of locally separable Moore spaces. This monograph should be of value to students, researchers, and specialists in the field of mathematics.

Inhalt

Front Cover 1
Set-Theoretic Topology 4
Copyright Page 5
Table of Contents 8
Dedication 6
CONTRIBUTORS 12
PREFACE 14
ACKNOWLEDGMENTS 16
CHAPTER 1. AN EXAMPLE CONCERNING THE PRESERVATION OF THE LINDELÖF PROPERTY IN PRODUCT SPACES 18
2. Some Lemmas 19
3. Proof of the Theorem 23
References 26
CHAPTER 2. CLOSED-COMPLETENESS IN SPACES WITH A QUASI-GdDIAGONAL 28
References 32
CHAPTER 3. CLOSED-COMPLETENESS IN SPACES WITH WEAK COVERING PROPERTIES 34
1. Introduction 34
2. Preliminaries 36
3. Weakly d.-refinable spaces 38
4. Weakly .-refinable spaces 47
5. Problems 57
References 59
CHAPTER 4. z-EMBEDDING IN ßX x ßY 64
Introduction 64
2. Pseudo-x1-compact spaces 69
3. One factor with a countable base 75
4. Appendix: The functors ooz and Ba 81
5. Appendix: z-embedding versus C*-embedding 84
References 87
CHAPTER 5. A NOTE ON d.-REFINABLE SPACES 90
1. Background 90
2. Introduction 91
3. Main Results 93
4. Addendum 96
References 97
CHAPTER 6. ON COUNTABLY COMPACT EXTENSIONS OF NORMAL LOCALLY COMPACT M-SPACES 98
References 105
CHAPTER 7. A CHARACTERIZATION OF CONTINUOUSLY SEMIMETRIZABLE SPACES 108
References 112
CHAPTER 8. DENSITY OF COMPACTIFICATIONS 114
0. Definitions and conventions 114
1. Introduction 115
2, The internal characterization 117
3. Examples 121
4. An example and Martin's Axiom restricted to separable spaces 123
References 126
CHAPTER 9. THE PIXLEY-ROY TOPOLOGY ON SPACES OF SUBSETS 128
1. Introduction 128
2. General results 131
3. Density of compactifications 134
4. A test space 140
5. A relation between the Pixley-Roy topology and the Vietoris topology 145
References 149
CHAPTER 10. SEPARATING CLOSED DISCRETE COLLECTIONS OF SINGULAR CARDINALITY 152
1. Example 153
2. Positive Results 154
3. Problems 156
References 156
CHAPTER 11. OPEN MAPPING THEORY 158
1. Introduction 158
2. Results involving open maps 160
3. Images and inverse images under open maps 172
5. Applications 198
References 202
CHAPTER 12. MOORE - CLOSED AND LOCALLY MOORE - CLOSED SPACES 210
References 233
CHAPTER 13. A CONSTRUCTION OF A QUASI-METRIC SOUSLIN SPACE WITH A POINT-COUNTABLE BASE 236
References 240
CHAPTER 14. ON GENERATING NON-ORTHOCOMPACT SPACES 242
References 252
CHAPTER 15. THE ALEXANDROFF-URYSOHN METRIZATION THEOREM REVISITED 256
1. Introduction 256
2. Definitions and survey of theorems 257
3. The construction 260
References 268
CHAPTER 16. THE G(m)-SPACES AND OTHER RELATED TOPICS 272
References 281
CHAPTER 17. STRONG S AND L SPACES UNDER MA 282
References 285
CHAPTER 18. A SELECTION-THEORETIC APPROACH TO CERTAIN EXTENSION THEOREMS 286
References 292
CHAPTER 19. SOME SURPRISING BASE PROPERTIES IN TOPOLOGY II 294
References 320
CHAPTER 20. SOME RESULTS FROM A-SPACES 324
References 329
CHAPTER 21. SOME OBSERVATIONS ON SEMICOMPLETABLE MOORE SPACES 330
References 340
CHAPTER 22. NORMALITY AND SEPARABILITY OF MOORE SPACES 342
1. Introduction 342
2. Normality of product spaces 344
3. Separable extensions of first countable and Moore spaces 346
4. Countably paracompact non-metrizahle Moore spaces 350
References 353
CHAPTER 23. ORTHOCOMPACTNESS IS NORMALITY IN FINITE PRODUCTS OF LOCALLY COMPACT LOTS'S 356
1. Notation and Terminology 356
2. A Locally Finite Closed Sum Theorem 357
3. Fat K'S 359
4. Locally Compact LOTS'S 363
References 365
CHAPTER 24. A REDUCTION OF THE HEREDITARILY SEPARABLE NON-LINDELÖF PROBLEM 366
References 368
CHAPTER 25. FIRST COUNTABLE SPACES WITH CALIBER .1 MAY OR MAY NOT BE SEPARABLE 370
References 374
CHAPTER 26. SOME EXAMPLES CONCERNING a-BOUNDED SPACES 376
1. Introduction 376
References 386
CHAPTER 27. NON-NORMAL SPACES 388
1. Introduction 388
2. Recent results 390
3. Construction of the examples 392
References 398
CHAPTER 28. COMPLETE MAPPINGS IN BASE OF COUNTABLE ORDER THEORY 400
1. Introduction 400
2. Invariance of primitive bases 402
3. Inductively open mappings 406
4. Primitive sequences 407
5. A mapping theorem for primitive sequences 410
6. A survey of base of countable order theory 413
7. Sequentially complete mappings 417
8. Forward invariance theorems 423
9. Inverse invariance theorems 425
References 428
CHAPTER 29. LOCALLY SEPARABLE MOORE SPACES 430
References 450

Produkt-Details

• Erscheint lt. Verlag: 10.5.2014
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Technik
• ISBN-10: 1-4832-6392-4 / 1483263924
• ISBN-13: 978-1-4832-6392-2 / 9781483263922