Topics in topology
type: Item
instance of: scholarly article
title: Topics in topology
zbMATH Open document ID: 0953.54001
published in: Lecture Notes in Mathematics
publication date: 1997-04-17T00:00:00Z
review text: The book aims to report new developments in the area of topology. It is divided into four chapters and each chapter is well-written with clear motivation. Chapter I `touches' the `Function space theory'; the word `touches' has been appropriately used by the author because he does not include the different types of topology, in vogue, in the theory of function spaces but gives the proof of some very important and interesting theorems of recent times. Starting with a theorem of Eberlein stating the `sequentially dense' nature of a subset \(A\) in \(C_p(X)\), where \(X\) is compact (\(C_p(X)\) denotes the space of all real-valued continuous functions with the topology of pointwise convergence), the author gives a proof of the Bourgain-Fremlin-Talagrand theorem about compact sets of Baire class-1 functions; Pták's approach to uniform convergence of convex combinations of elements of a fixed sequence of continuous functions which pointwise accumulates to some continuous function has been treated to prove Namioka's theorem about separate versus joint continuity in product spaces. In Chapter II a semigroup structure on the Čech-Stone compactifications of the integers is studied with an eye to applying it for giving a topological proof for the famous Van der Waerden theorem. Some recent applications of topological methods to combinatorics and number theory such as the works of S. Glazer, N. Hindman and H. Furstenberg are presented. In Chapter III the author concentrates on the structure theory of compact, locally compact and compactly generated groups; however, the motivation is not to give a complete overview of the theory but to present a proof of the famous theorem that compact groups are dyadic; references are also made to some far reaching extensions of this theorem; further, the author gives a proof of Pestov's theorem that every \(\sigma\)-compact group can be embedded into a compactly generated one and Tkachenko's result that \(\sigma\)-compact groups are Souslin. I…
reviewed by: Sobhakar Ganguly
Mathematics Subject Classification ID: 54-02
Mathematics Subject Classification ID: 54H11
Mathematics Subject Classification ID: 22D05
Mathematics Subject Classification ID: 26A21
Mathematics Subject Classification ID: 28A60
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 54E50
Mathematics Subject Classification ID: 54C35
zbMATH DE Number: 1001281
zbMATH Keywords: Bourgain-Fremlin-Talagrand theorem
zbMATH Keywords: Namioka's theorem
zbMATH Keywords: Van der Waerden theorem
zbMATH Keywords: Pestov's theorem
zbMATH Keywords: exponential space
zbMATH Keywords: accumulation orders
author: Stevo Todorčević
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bfb0096295
OpenAlex ID: W4244079350
DOI: 10.1007/BFB0096295
description: scientific article; zbMATH DE number 1001281