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Topological function spaces. Transl. from the Russian by R. A. M. Hoksbergen. (English) Zbl 0758.46026

The present volume studies the space \(C_ p(X)\) of all real valued continuous functions on a topological space \(X\) in the topology of pointwise convergence. The topological properties of \(C_ p(X)\) and its subspaces are very important in many areas, mainly in functional analysis and general topology.
The book has five chapters. In Chapter 0 we can find general questions about \(C_ p(X)\), certain notions from general topology, terminology and notations which are useful further. Chapter 0 considers \(C_ p(X)\) as an object of topological algebra and concludes with Nagata’s theorem, which shows the main advantage of the topology of pointwise convergence over the compact-open topology and other topologies: “Tikhonov spaces \(X\) and \(Y\) are homeomorphic if and only if the topological rings \(C_ p(X)\) and \(C_ p(Y)\) are topologically isomorphic”.
Chapter I solves a fundamental question: How are the properties of \(X\) and \(C_ p(X)\) related? Special interest is attached to duality properties of \(X\) and \(C_ p(X)\), i.e. properties characterizing each other. Here we can find answers to the questions: when \(C_ p(X)\) is \(\sigma\)-compact, Čech complete, paracompact, ….
Chapter II shows for which \(X\) the space \(C_ p(X)\) has countable tightness. In this part also a criterion for \(C_ p(X)\) to be a Fréchet-Urysohn space in terms of the topology of \(X\) is given. Using this criterion it is established that if \(C_ p(X)\) is a Fréchet- Urysohn space, then \((C_ p(X))^{\aleph_ 0}\) is such.
The aim of Chapter III is to deepen the understanding of the peculiarities of the topological structure of function spaces over compacta. It was shown in the previous parts that such spaces are monolithic and have countable tightness. For \(X\) compact, the conditions \(X\) is scattered and \(C_ p(X)\) is Fréchet-Urysohn space are equivalent. This chapter contains also the following successful generalization of a well-known theorem of Grothendieck due to Asanov and Velichko: If a space \(X\) is countably compact, then the closure \(\overline {F}\) in \(C_ p(X)\) of any set \(F\) bounded in \(C_ p(X)\) is a compactum in \(C_ p(X)\).
In Chapter IV is proved for a large class of compacta that the function spaces over them have countable Lindelöf number.
This volume contains many new and original results and also many problems unsolved up to now. The book concludes with a bibliography.

MSC:

46E10 Topological linear spaces of continuous, differentiable or analytic functions
46-02 Research exposition (monographs, survey articles) pertaining to functional analysis
54C35 Function spaces in general topology
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
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