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Topological Groups and Related Structures / by Alexander Arhangel’skii, Mikhail Tkachenko
Contributor(s): Resource type: Ressourcentyp: Buch (Online)Book (Online)Language: English Series: Atlantis Studies in Mathematics ; 1 | SpringerLink BücherPublisher: Paris : Atlantis Press, 2008Description: Online-Ressource (XIV, 781p, digital)ISBN:- 9789491216350
- 512.2
- 514
- QA174-183
Contents:
Summary: to Topological Groups and Semigroups -- Right Topological and Semitopological Groups -- Topological groups: Basic constructions -- Some Special Classes of Topological Groups -- Cardinal Invariants of Topological Groups -- Moscow Topological Groups and Completions of Groups -- Free Topological Groups -- R-Factorizable Topological Groups -- Compactness and its Generalizations in Topological Groups -- Actions of Topological Groups on Topological Spaces.Summary: Algebraandtopology,thetwofundamentaldomainsofmathematics,playcomplem- tary roles. Topology studies continuity and convergence and provides a general framework to study the concept of a limit. Much of topology is devoted to handling in?nite sets and in?nity itself; the methods developed are qualitative and, in a certain sense, irrational. - gebra studies all kinds of operations and provides a basis for algorithms and calculations. Very often, the methods here are ?nitistic in nature. Because of this difference in nature, algebra and topology have a strong tendency to develop independently, not in direct contact with each other. However, in applications, in higher level domains of mathematics, such as functional analysis, dynamical systems, representation theory, and others, topology and algebra come in contact most naturally. Many of the most important objects of mathematics represent a blend of algebraic and of topologicalstructures. Topologicalfunctionspacesandlineartopologicalspacesingeneral, topological groups and topological ?elds, transformation groups, topological lattices are objects of this kind. Very often an algebraic structure and a topology come naturally together; this is the case when they are both determined by the nature of the elements of the set considered (a group of transformations is a typical example). The rules that describe the relationship between a topology and an algebraic operation are almost always transparentandnatural—theoperationhastobecontinuous,jointlyorseparately.PPN: PPN: 1651281378Package identifier: Produktsigel: ZDB-2-SEB | ZDB-2-SXMS | ZDB-2-SMA
Topological Groups and Related Structures; Foreword; Contents; Chapter 1: Introduction to Topological Groups and Semigroups ; 1.1. Some algebraic concepts; 1.2. Groups and semigroups with topologies; 1.3. Neighbourhoods of the identity in topological groups and semigroups; 1.4. Open sets, closures, connected sets and compact sets; 1.5. Quotients of topological groups; 1.6. Products, Σ-products, and σ-products; 1.7. Factorization theorems; 1.8. Uniformities on topological groups; 1.9. Markov's theorem; 1.10. Historical comments to Chapter 1
Chapter 2: Right Topological and Semitopological Groups2.1. From discrete semigroups to compact semigroups; 2.2. Idempotents in compact semigroups; 2.3. Joint continuity and continuity of the inverse in semitopological groups; 2.4. Pseudocompact semitopological groups; 2.5. Cancellative topological semigroups; 2.6. Historical comments to Chapter 2; Chapter 3: Topological groups: Basic constructions; 3.1. Locally compact topological groups; 3.2. Quotients with respect to locally compact subgroups; 3.3. Prenorms on topological groups, metrization; 3.4. ω-narrow and ω-balanced topological groups
3.5. Groups of isometries and groups of homeomorphisms3.6. Ra˘ıkov completion of a topological group; 3.7. Precompact groups and precompact sets; 3.8. Embeddings into connected, locally connected groups; 3.9. Historical comments to Chapter 3; Chapter 4: Some Special Classes of Topological Groups; 4.1. Ivanovskij-Kuz'minov Theorem; 4.2. Embedding Dω1 in a non-metrizable compact group; 4.3. Cˇ ech-complete and feathered topological groups; 4.4. P-groups; 4.5. Extremally disconnected topological and quasitopological groups; 4.6. Perfect mappings and topological groups
4.7. Some convergence phenomena in topological groups4.8. Historical comments to Chapter 4; Chapter 5: Cardinal Invariants of Topological Groups; 5.1. More on embeddings in products of topological groups; 5.2. Some basic cardinal invariants of topological groups; 5.3. Lindel¨of Σ-groups and Nagami number; 5.4. Cellularity and weak precalibres; 5.5. o-tightness in topological groups; 5.6. Steady and stable topological groups; 5.7. Cardinal invariants in paratopological and semitopological groups; 5.8. Historical comments to Chapter 5
Chapter 6: Moscow Topological Groups and Completions of Groups6.1. Moscow spaces and C-embeddings; 6.2. Moscow spaces, P-spaces, and extremal disconnectedness; 6.3. Products and mappings of Moscow spaces; 6.4. The breadth of the class of Moscow groups; 6.5. When the Dieudonn´e completion of a topological group is a group?; 6.6. Pseudocompact groups and their completions; 6.7. Moscow groups and the formula υ(X × Y) = υX × υY; 6.8. Subgroups of Moscow groups; 6.9. Pointwise pseudocompact and feathered groups; 6.10. Bounded and C-compact sets; 6.11. Historical comments to Chapter 6
Chapter 7: Free Topological Groups
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