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