This book is the second volume of the Handbook of the History of General Topology. As was the case for the first volume, the contributions contained in it concern either individual topologists, specific schools of topology, specific periods of development, specific topics or a combination of these. The second volume focuses on the work of famous topologists, such as W. Sierpinski, K. Kuratowski (both by R. Engelkind), S. Mazurkiewicz (by R. Pol) and R.G. Bing (by M. Starbird). Furthermore, it contains articles covering Uniform, Proximinal and Nearness Concepts in Topology (by H.L. Bentley, H. Herrlich, M. Husek), Hausdorff Compactifications (by R.E. Chandler, G. Faulkner), Continua Theory (by J.J. Charatonik), Generalized Metrizable Spaces (by R.E. Hodel), Minimal Hausdorff Spaces and Maximally Connected Spaces (by J.R. Porter, R.M. Stephenson Jr.), Orderable Spaces (by S. Purisch), Developable Spaces (by S.D. Shore) and The Alexandroff-Sorgenfrey Line (by D.E. Cameron). Together with the first volume and the forthcoming volume(s) this work on the history of topology, in all its aspects, is unique, and presents important views and insights into the problems and development of topological theories and applications of topological concepts, and into the life and work of topologists. As such it will encourage not only further study in the history of the subject, but also further mathematical research in the field. It is an invaluable tool for topology researchers and topology teachers throughout the mathematical world.
Author(s): C.E. Aull, R. Lowen
Edition: 1
Publisher: Springer
Year: 1998
Language: English
Pages: 419
Cover......Page 1
Title Page......Page 3
Copyright Page......Page 4
Contents ......Page 5
Introduction ......Page 7
Waclaw Sierpinski (1882-1969) - His Life and Work in Topology ......Page 11
1 Biographical sketch ......Page 13
2 Work in topology ......Page 15
2.1 Curves and continua ......Page 16
2.2 Highly disconnected sets and dimension theory ......Page 17
2.3 General topological properties ......Page 20
2.4 Descriptive set theory ......Page 21
The Works of Stefan Mazurkiewicz in Topology ......Page 27
1 Introduction ......Page 29
2 Continuum Theory ......Page 31
3 Dimension theory ......Page 33
4 Descriptive set theory ......Page 36
5 Miscellaneous results ......Page 38
Kazimierz Kuratowski (1896-1980) - His Life and Work in Topology ......Page 43
1 Biographical sketch ......Page 45
2.1 Continua ......Page 48
2.2 Topology of the plane and of Euclidean n-space ......Page 51
2.3 Dimension theory ......Page 53
2.4 Descriptive set theory ......Page 55
2.5 General topological properties ......Page 56
2.6 Hyperspaces and selectors. Function spaces ......Page 58
R.H. Bing's Human and Mathematical Vitality ......Page 65
From Developments to Developable Spaces ......Page 79
1 1900-19 10: Primordial Beginnings ......Page 82
1.1 E.H. Moore's Notion of Development for an Abstract Class ......Page 84
2 1910-1915: Refining the Neighborhood Concept ......Page 89
2.1 The Precursory Contributions of Hilbert and Veblen ......Page 90
2.2 The Early Contributions of Frechet and Riesz ......Page 92
2.3 The Contribution of Hedrick ......Page 96
2.4 The Contribution of Hildebrandt ......Page 101
2.5 The Contribution of Root ......Page 103
2.6 The Contribution of Hausdorff ......Page 109
3 1916-1923: The Influence of "The Metrization Problem" ......Page 111
3.1 Chittenden's First Metrization Theorem ......Page 113
3.2 Frechet Poses "The Metrization Problem" ......Page 115
3.3 The Joint Work of Chittenden and Pitcher ......Page 118
3.4 Alexandroff and Urysohn's Metrization Theorem ......Page 129
4.1 Chittenden's Outline for General Topology ......Page 133
4.2 Contributions of the 1930's ......Page 141
4.3 Bing's Developable Spaces ......Page 145
A History of Generalized Metrizable Spaces ......Page 153
1 Developable, semi-metrizable, and quasi-metrizable spaces (early history) ......Page 157
2 Nine metrization theorems: 1923-1957 ......Page 159
3 The role of covering properties in generalized metrizable spaces ......Page 162
4 Unification with compactness: p-, M-, wL-, E-, and a-spaces ......Page 164
5 A-, stratifiable, and Nagata spaces ......Page 169
6 Base of countable order, 9-base, and quasi-developable spaces ......Page 170
7 g-functions: a unified approach ......Page 171
8 Nets and a-spaces ......Page 173
9 Ga-diagonal, point-countable base, point-countable separating open cover ......Page 174
10 Semi-metrizable, symmetrizable, and semi-stratifiable spaces ......Page 177
11 Quasi-metrizable and -y-spaces ......Page 179
12 Concluding remarks ......Page 182
The Historical Development of Uniform, Proximal, and Nearness Concepts in Topology ......Page 189
1.1 Uniform concepts before 1900 ......Page 191
1.2 Uniform concepts till 1937 ......Page 194
1.3 From A.Weil to J.R. Isbell ......Page 201
1.4 Uniform spaces in the 60's and 70's ......Page 206
1.5 Proximity spaces ......Page 211
2 Generalized Uniform Structures ......Page 215
2.1 Preamble ......Page 216
2.2 Merotopic spaces and filter-merotopic spaces ......Page 218
2.3 Cauchy spaces ......Page 221
2.4 Nearness spaces ......Page 222
2.5 Further developments about categories of merotopic spaces ......Page 224
Hausdorff Compactifications: A Retrospective ......Page 243
1 Compactifications: Early Efforts ......Page 248
2 Modern Notation and Conventions ......Page 253
3 The Structure of K(X) ......Page 254
4 Special Properties ......Page 256
5 Singular Compactifications ......Page 259
6 \beta N ......Page 261
7 Alternative Constructions of \beta N ......Page 265
8 The Axiom of Choice ......Page 266
9 Wallman-Frink Compactifications ......Page 267
10 Secondary Sources ......Page 269
11 Research Levels Since 1940 ......Page 270
12 Appendix: On the Origin of "Stone-tech" ......Page 271
13 Appendix: Lubben's discovery of the "Stone-tech" Compactification ......Page 272
Minimal Hausdorff Spaces - Then and Now ......Page 281
1 Minimal Spaces ......Page 284
2 Katetov Spaces ......Page 290
3 Problems ......Page 294
A History of Results on Orderability and Suborderability ......Page 301
History of Continuum Theory ......Page 315
2 Basic concepts ......Page 317
3 The Jordan Curve Theorem and the concept of a curve ......Page 319
4 Local connectedness; plane continua ......Page 321
5 Indecomposability ......Page 328
6 Irreducible continua; decompositions ......Page 333
7 Hereditary indecomposability; P-like continua ......Page 340
8 Homogeneity ......Page 345
9 Mapping properties - families of continua ......Page 351
10 Special mappings ......Page 354
11 Fixed point theory ......Page 358
12 Hyperspaces ......Page 362
13 Final remarks ......Page 365
Why I Study the History of Mathematics ......Page 399
The Alexandroff-Sorgenfrey Line ......Page 403
The Flowering of General Topology in Japan - Correction ......Page 409
Index ......Page 411
