A \(C_p\)-theory problem book. Functional equivalencies
type: Item
instance of: scholarly article
title: A \(C_p\)-theory problem book. Functional equivalencies
zbMATH Open document ID: 1354.54001
DOI: 10.1007/978-3-319-24385-6
author: Vladimir Tkachuk
published in: Problem Books in Mathematics
publication date: 2015-10-08T00:00:00Z
review text: 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 ea…
Mathematics Subject Classification ID: 54-02
Mathematics Subject Classification ID: 00A07
Mathematics Subject Classification ID: 46E99
Mathematics Subject Classification ID: 54-01
Mathematics Subject Classification ID: 54C35
zbMATH DE Number: 6492281
zbMATH Keywords: \(C_p\)-theory
zbMATH Keywords: \(t\)-equivalence
zbMATH Keywords: \(l\)-equivalence
zbMATH Keywords: \(u\)-equivalence
zbMATH Keywords: cardinal invariants
reviewed by: Ljubiša D. R. Kočinac
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-319-24385-6
OpenAlex ID: W4248639466
description: scientific article; zbMATH DE number 6492281