Franklin

Ultrafilters and topologies on groups [electronic resource] / Yevhen G. Zelenyuk.

Author/Creator:
Zelenyuk, Yevhen G.
Publication:
Berlin ; New York : De Gruyter, c2011.
Format/Description:
Book
1 online resource (228 p.)
Series:
De Gruyter expositions in mathematics ; 50.
De Gruyter expositions in mathematics, 0938-6572 ; 50
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Subjects:
Topological group theory.
Ultrafilters (Mathematics).
Form/Genre:
Electronic books.
Language:
English
Summary:
This book presents the relationship between ultrafilters and topologies on groups. It shows how ultrafilters are used in constructing topologies on groups with extremal properties and how topologies on groups serve in deriving algebraic results about ultrafilters. The contents of the book fall naturally into three parts. The first, comprising Chapters 1 through 5, introduces to topological groups and ultrafilters insofar as the semigroup operation on ultrafilters is not required. Constructions of some important topological groups are given. In particular, that of an extremally disconnected topological group based on a Ramsey ultrafilter. Also one shows that every infinite group admits a nondiscrete zero-dimensional topology in which all translations and the inversion are continuous. In the second part, Chapters 6 through 9, the Stone-Cêch compactification βG of a discrete group G is studied. For this, a special technique based on the concepts of a local left group and a local homomorphism is developed. One proves that if G is a countable torsion free group, then βG contains no nontrivial finite groups. Also the ideal structure of βG is investigated. In particular, one shows that for every infinite Abelian group G, βG contains 22|G| minimal right ideals. In the third part, using the semigroup βG, almost maximal topological and left topological groups are constructed and their ultrafilter semigroups are examined. Projectives in the category of finite semigroups are characterized. Also one shows that every infinite Abelian group with finitely many elements of order 2 is absolutely ω-resolvable, and consequently, can be partitioned into ω subsets such that every coset modulo infinite subgroup meets each subset of the partition. The book concludes with a list of open problems in the field. Some familiarity with set theory, algebra and topology is presupposed. But in general, the book is almost self-contained. It is aimed at graduate students and researchers working in topological algebra and adjacent areas.
Contents:
Frontmatter
Preface
Contents
1 Topological Groups
2 Ultrafilters
3 Topological Spaces with Extremal Properties
4 Left Invariant Topologies and Strongly Discrete Filters
5 Topological Groups with Extremal Properties
6 The Semigroup βS
7 Ultrafilter Semigroups
8 Finite Groups in βG
9 Ideal Structure of βG
10 Almost Maximal Topological Groups
11 Almost Maximal Spaces
12 Resolvability
13 Open Problems
Bibliography
Index
Notes:
Description based upon print version of record.
Includes bibliographical references and index.
ISBN:
1-283-16472-8
9786613164728
3-11-021322-2
OCLC:
723945447
Publisher Number:
10.1515/9783110213225 doi