Topological groups and related structures / Alexander Arhangel'skii, Mikhail Tkachenko.

Format:

Book

Author/Creator:

Arkhangelʹskiĭ, A. V.

Contributor:

Tkachenko, Mikhail.

Alumni and Friends Memorial Book Fund.

Series:

Atlantis studies in mathematics ; v. 1.

Atlantis studies in mathematics ; v. 1

Language:

English

Subjects (All):

Topological groups.

Physical Description:

xiv, 781 pages ; 26 cm.

Place of Publication:

Amsterdam : Atlantis Press ; [Singapore] : World Scientific, [2008]

Summary:

This book presents a large amount of material, both classic and recent (on occasion, unpublished) about the relations of Algebra and Topology. It therefore belongs to the area called Topological Algebra. More specifically, the objects of the study are subtle and sometimes unexpected phenomena that occur when the continuity meets and properly feeds an algebraic operation. Such a combination gives rise to many classic structures, including topological groups and semigroups, paratopological groups, etc. Special emphasis is given to tracing the influence of compactness and its generalizations on the properties of an algebraic operation, causing on occasion the automatic continuity of the operation. The main scope of the book, however, is outside of the locally compact structures, thus distinguishing the monograph from a series of more traditional textbooks.The book is unique in that it presents very important material, dispersed in hundreds of research articles, not covered by any monograph in existence. The reader is gently introduced to an amazing world at the interface of Algebra, Topology, and Set Theory. He/she will find that the way to the frontier of the knowledge is quite short -- almost every section of the book contains several intriguing open problems whose solutions can contribute significantly to the area.

Contents:

Chapter 1 Introduction to Topological Groups and Semigroups 1

1.1 Some algebraic concepts 1

1.2 Groups and semigroups with topologies 12

1.3 Neighbourhoods of the identity in topological groups and semigroups 18

1.4 Open sets, closures, connected sets and compact sets 26

1.5 Quotients of topological groups 37

1.6 Products, [Sigma]-products, and [sigma]-products 46

1.7 Factorization theorems 62

1.8 Uniformities on topological groups 66

1.9 Markov's theorem 81

1.10 Historical comments to Chapter 1 87

Chapter 2 Right Topological and Semitopological Groups 90

2.1 From discrete semigroups to compact semigroups 91

2.2 Idempotents in compact semigroups 97

2.3 Joint continuity and continuity of the inverse in semitopological groups 109

2.4 Pseudocompact semitopological groups 120

2.5 Cancellative topological semigroups 129

2.6 Historical comments to Chapter 2 131

Chapter 3 Topological groups: Basic constructions 134

3.1 Locally compact topological groups 134

3.2 Quotients with respect to locally compact subgroups 147

3.3 Prenorms on topological groups, metrization 151

3.4 [omega]-narrow and [omega]-balanced topological groups 162

3.5 Groups of isometries and groups of homeomorphisms 173

3.6 Raikov completion of a topological group 181

3.7 Precompact groups and precompact sets 193

3.8 Embeddings into connected, locally connected groups 202

3.9 Historical comments to Chapter 3 212

Chapter 4 Some Special Classes of Topological Groups 216

4.1 Ivanovskij-Kuz'minov Theorem 217

4.2 Embedding D[superscript omega 1] in a non-metrizable compact group 226

4.3 Cech-complete and feathered topological groups 230

4.4 P-groups 249

4.5 Extremally disconnected topological and quasitopological groups 255

4.6 Perfect mappings and topological groups 264

4.7 Some convergence phenomena in topological groups 273

4.8 Historical comments to Chapter 4 282

Chapter 5 Cardinal Invariants of Topological Groups 285

5.1 More on embeddings in products of topological groups 286

5.2 Some basic cardinal invariants of topological groups 296

5.3 Lindelof [Sigma]-groups and Nagami number 303

5.4 Cellularity and weak precalibres 316

5.5 o-tightness in topological groups 323

5.6 Steady and stable topological groups 329

5.7 Cardinal invariants in paratopological and semitopological groups 336

5.8 Historical comments to Chapter 5 343

Chapter 6 Moscow Topological Groups and Completions of Groups 345

6.1 Moscow spaces and C-embeddings 346

6.2 Moscow spaces, P-spaces, and extremal disconnectedness 351

6.3 Products and mappings of Moscow spaces 354

6.4 The breadth of the class of Moscow groups 359

6.5 When the Dieudonne completion of a topological group is a group? 366

6.6 Pseudocompact groups and their completions 374

6.7 Moscow groups and the formula v(X x Y) = vX x vY 378

6.8 Subgroups of Moscow groups 385

6.9 Pointwise pseudocompact and feathered groups 388

6.10 Bounded and C-compact sets 395

6.11 Historical comments to Chapter 6 407

Chapter 7 Free Topological Groups 409

7.1 Definition and basic properties 409

7.2 Extending pseudometrics from X to F(X) 424

7.3 Extension of metrizable groups by compact groups 443

7.4 Direct limit property and completeness 446

7.5 Precompact and bounded sets in free groups 453

7.6 Free topological groups on metrizable spaces 456

7.7 Nummela-Pestov theorem 476

7.8 The direct limit property and countable compactness 482

7.9 Completeness of free Abelian topological groups 490

7.10 M-equivalent spaces 499

7.11 Historical comments to Chapter 7 511

Chapter 8 R-Factorizable Topological Groups 515

8.1 Basic properties 515

8.2 Subgroups of R-factorizable groups. Embeddings 526

8.3 Dieudonne completion of R-factorizable groups 532

8.4 Homomorphic images of R-factorizable groups 535

8.5 Products with a compact factor and m-factorizability 538

8.6 R-factorizability of P-groups 550

8.7 Factorizable groups and projectively Moscow groups 562

8.8 Zero-dimensionality of R-factorizable groups 565

8.9 Historical comments to Chapter 8 569

Chapter 9 Compactness and its Generalizations in Topological Groups 571

9.1 Krein-Milman Theorem 571

9.2 Gel'fand-Mazur Theorem 576

9.3 Invariant integral on a compact group 581

9.4 Existence of non-trivial continuous characters on compact Abelian groups 594

9.5 Pontryagin-van Kampen duality theory for discrete and for compact groups 604

9.6 Some applications of Pontryagin-van Kampen's duality 610

9.7 Non-trivial characters on locally compact Abelian groups 621

9.8 Varopoulos' theorem: the Abelian case 624

9.9 Bohr topology on discrete Abelian groups 633

9.10 Bounded sets in extensions of groups 663

9.11 Pseudocompact group topologies on Abelian groups 666

9.12 Countably compact topologies on Abelian groups 675

9.13 Historical comments to Chapter 9 692

Chapter 10 Actions of Topological Groups on Topological Spaces 697

10.1 Dugundji spaces and 0-soft mappings 697

10.2 Continuous action of topological groups on spaces 708

10.3 Uspenskij's theorem on continuous transitive actions of [omega]-narrow groups on compacta 716

10.4 Continuous actions of compact groups and some classes of spaces 723

10.5 Historical comments to Chapter 10 732.

Notes:

Includes bibliographical references (pages 735-753) and indexes.

Local Notes:

Acquired for the Penn Libraries with assistance from the Alumni and Friends Memorial Book Fund.

ISBN:

9789078677062

9078677066

OCLC:

191658547

Publisher Number:

99937462804