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Research and Exposition in Mathematics -- Volume 31

   Enlarged Picture

Gábor Lukács

Compact-like Topological Groups

xiv+166 pages, soft cover, ISBN 978-3-88538-231-7, EUR 28.00, 2009

The class of compact topological groups is perhaps the most understood part of topological groups. This monograph deals with larger classes defined by properties shared by compact groups. For instance, if G is a compact topological group, then for every group H, the image of every closed subgroup of G × H under the second projection is closed in H. Topological groups with this property are called c-compact.

Is every c-compact topological group compact?

Chapter 1 is an elementary introduction to topological groups.
Chapter 2 presents cardinal invariants, a more specialized aspect of topological group theory.
Chapter 3 focuses on two compactness-like properties, namely, precompactness and minimality.
Chapter 4 relates to the main question by proving compactness theorems (that is, results of the type: if G is c-compact and has an additional property, then G is compact), open mapping theorems, and reduction theorems (reducing the general question to smaller classes of groups).

List of Contents:

  Preface v
  Introduction vii
  Acknowledgments xi
  Chapter 1: Preliminaries  
1.1 Neighborhoods of the identity 2
1.2 Locally compact groups 15
1.3 Completeness 22
1.4 Notes 28
Chapter 2: Cardinal invariants of topological groups  
2.1 The Gτ-topology and τ-representable groups 32
2.2 τ-balanced groups 36
2.3 τ-precompact groups 42
2.4 Closed subgroups of products 47
2.5 Notes 58
Chapter 3: Minimal and totally minimal groups  
3.1 Precompactness 61
3.2 Minimality and total minimality 67
3.3 Extensions and products 77
3.4 Notes 84
Chapter 4: From c-compactness to sequential h-completeness  
4.1 Basics 90
4.2 Characterizations using special filters 99
4.3 Open mapping and structure theorems 113
4.4 Compactness theorems 121
4.5 Special cases: Locally compact and discrete groups 131
4.6 Notes and open problems 135
Appendix: History of the problem of c-compactness 139
List of symbols 147
Bibliography 149