Topics in topology
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. It may especially be mentioned that a considerable amount of information about compact and compactly generated groups is given in the exercises. Chapter IV deals with the exponential space \(\exp(X)\) of a topological space \(X\) when it is compact metric. The main concentration is however in the exposition of the theory of accumulation orders and spectra of compact metric zero-dimensional spaces developed by Marjanović in order to classify the space \(\exp(X)\) in the class of metric compacta. Throughout the text a number of exercises are inserted which are chosen as corollaries to the material presented in the text and some others are there which inform the reader about the material which is not presented here.
scientific article; zbMATH DE number 1001281