Chain Conditions in Topology

Author: W. W. Comfort,William Wistar Comfort,S. Negrepontis

Publisher: Cambridge University Press

ISBN: 0521234875

Category: Mathematics

Page: 300

View: 5653


A chain condition is a property, typically involving considerations of cardinality, of the family of open subsets of a topological space. (Sample questions: (a) How large a fmily of pairwise disjoint open sets does the space admit? (b) From an uncountable family of open sets, can one always extract an uncountable subfamily with the finite intersection property. This monograph, which is partly fresh research and partly expository (in the sense that the authors co-ordinate and unify disparate results obtained in several different countries over a period of several decades) is devoted to the systematic use of infinitary combinatorial methods in topology to obtain results concerning chain conditions. The combinatorial tools developed by P. Erdös and the Hungarian school, by Erdös and Rado in the 1960s and by the Soviet mathematician Shanin in the 1940s, are adequate to handle many natural questions concerning chain conditions in product spaces.

Surveys in General Topology

Author: George M. Reed

Publisher: Academic Press

ISBN: 148326386X

Category: Mathematics

Page: 572

View: 4131


Surveys in General Topology presents topics relating to general topology ranging from closed mappings and ultrafilters to covering and separation properties of box products. Ordered topological spaces and the use of combinatorial techniques in functional analysis are also considered, along with product spaces and weakly compact subsets of Banach spaces. Applications of stationary sets in topology are presented as well. Comprised of 15 chapters, this volume begins with an analysis of some of the techniques and results in the area of closed mappings, followed by a discussion on the theory of ultrafilters. The reader is then introduced to the question of when a box product of compact spaces is paracompact, and how badly a box product of compact or metrizable spaces can fail to be normal. Subsequent chapters focus on the transfinite dimension; the properties of metacompactness, submetacompactness, and subparacompactness; the dimension of ordered topological spaces; the use of combinatorial techniques for the treatment and solution of fundamental problems in functional analysis, particularly in the isomorphic theory of Banach spaces; and order-theoretic base axioms. This monograph will be of significant value both to researchers in general topology and to mathematicians outside the field who wish an overview of current topics and techniques.

Recent Progress in General Topology II

Author: Miroslav Hušek,J. van Mill

Publisher: Elsevier

ISBN: 0444509801

Category: Mathematics

Page: 638

View: 6597


The book presents surveys describing recent developments in most of the primary subfields of General Topology and its applications to Algebra and Analysis during the last decade. It follows freely the previous edition (North Holland, 1992), Open Problems in Topology (North Holland, 1990) and Handbook of Set-Theoretic Topology (North Holland, 1984). The book was prepared in connection with the Prague Topological Symposium, held in 2001. During the last 10 years the focus in General Topology changed and therefore the selection of topics differs slightly from those chosen in 1992. The following areas experienced significant developments: Topological Groups, Function Spaces, Dimension Theory, Hyperspaces, Selections, Geometric Topology (including Infinite-Dimensional Topology and the Geometry of Banach Spaces). Of course, not every important topic could be included in this book. Except surveys, the book contains several historical essays written by such eminent topologists as: R.D. Anderson, W.W. Comfort, M. Henriksen, S. Mardeŝić, J. Nagata, M.E. Rudin, J.M. Smirnov (several reminiscences of L. Vietoris are added). In addition to extensive author and subject indexes, a list of all problems and questions posed in this book are added. List of all authors of surveys: A. Arhangel'skii, J. Baker and K. Kunen, H. Bennett and D. Lutzer, J. Dijkstra and J. van Mill, A. Dow, E. Glasner, G. Godefroy, G. Gruenhage, N. Hindman and D. Strauss, L. Hola and J. Pelant, K. Kawamura, H.-P. Kuenzi, W. Marciszewski, K. Martin and M. Mislove and M. Reed, R. Pol and H. Torunczyk, D. Repovs and P. Semenov, D. Shakhmatov, S. Solecki, M. Tkachenko.

A Cp-Theory Problem Book

Compactness in Function Spaces

Author: Vladimir V. Tkachuk

Publisher: Springer

ISBN: 3319160923

Category: Mathematics

Page: 524

View: 1902


This third volume in Vladimir Tkachuk's series on Cp-theory problems applies all modern methods of Cp-theory to study compactness-like properties in function spaces and introduces the reader to the theory of compact spaces widely used in Functional Analysis. The text is designed to bring a dedicated reader from basic topological principles to the frontiers of modern research covering a wide variety of topics in Cp-theory and general topology at the professional level. The first volume, Topological and Function Spaces © 2011, provided an introduction from scratch to Cp-theory and general topology, preparing the reader for a professional understanding of Cp-theory in the last section of its main text. The second volume, Special Features of Function Spaces © 2014, continued from the first, giving reasonably complete coverage of Cp-theory, systematically introducing each of the major topics and providing 500 carefully selected problems and exercises with complete solutions. This third volume is self-contained and works in tandem with the other two, containing five hundred carefully selected problems and solutions. It can also be considered as an introduction to advanced set theory and descriptive set theory, presenting diverse topics of the theory of function spaces with the topology of point wise convergence, or Cp-theory which exists at the intersection of topological algebra, functional analysis and general topology.