A \(C_p\)-theory problem book. Functional equivalencies
This is the last volume of the four-volume set by the author [Topological and function spaces. Berlin: Springer (2011; Zbl 1222.54002); Special features of function spaces. Cham: Springer (2014; Zbl 1303.54006); Compactness in function spaces. Cham: Springer (2015; Zbl 1325.54001)]. The book presents a systematic exposition of functional equivalences of Tychonoff spaces and provides many major results and methods in the area of \(C_p\)-theory related to preservation or non-preservation of topological properties by different sorts of homeomorphisms between function spaces. This book, as the previous three, is intended for graduate and postgraduate students and for all those who are interested in learning more advanced results and investigation in this dynamic area of topology that has various applications, especially in functional analysis. This volume is divided into four chapters (1. Properties preserved by homeomorphisms of function spaces, 2. Solutions of problems 001--500, 3. Bonus results: some hidden statements, 4. Open problems) and ends by a bibliography containing 800 items. The first two chapters contain the main material of the book. Chapter 1 contains statements of 500 selected problems (divided into five sections, each with 100 problems), going up to deep major results regarding \(t\)-equivalence, \(l\)-equivalence and \(u\)-equivalence. For example, among the problems are the classical Kolmogorov solution of Hilbert's Problem 13, Okunev's method of constructing \(l\)-equivalent pairs of spaces, the Velichko-Bouziad result on \(l\)-invariance of the Lindelöf number, the result of Gul'ko and Khmyleva that compactness is not \(t\)-invariant, Okunev's result on preservation of \(\sigma\)-compactness by \(t\)-equivalence, Marciszewski's result that the unit segment \([0,1]\) and its countable power are not \(t\)-equivalent, Gul'ko's result on \(u\)-invariance of the dimension dim and Pestov's result on \(l\)-invariance of dim, and so on. The beginning of each section supplies definitions and basis results used in that section, and the section ends with bibliographic comments. All problems from Chapter 1 are completely solved in Chapter 2. There is a lot of additional bonus information including more that 350 notes, called here facts, supplied also with detailed proofs. Some of these notes are important known theorems that extend the main text and provide important bibliographical and historical background. Chapter 3 contains 80 hidden topological results that have been used without special mentioning in the solutions of the problems in Chapter 2. Chapter 4 provides 100 open problems. Some of them are quite difficult and probably need some new sophisticated methods for their solution. Breadth, depth and clarity of the exposition and the informative style make this book (and the entire set of four volumes) a valuable reference source for several areas of general topology that can help to inform and direct future independent research.
scientific article; zbMATH DE number 6492281