Pseudocompact topological spaces. A survey of classic and new results with open problems
type: Item
instance of: scholarly article
title: Pseudocompact topological spaces. A survey of classic and new results with open problems
zbMATH Open document ID: 1411.54002
DOI: 10.1007/978-3-319-91680-4
published in: Developments in Mathematics
publication date: 2018-06-29T00:00:00Z
review text: In [Trans. Am. Math. Soc. 64, 45--99 (1948; Zbl 0032.28603)] \textit{E. Hewitt} extensively studied properties of the rings \(C(X)\) and \(C^*(X)\) of continuous and bounded continuous real-valued functions defined on a Tychonoff space \(X\), and partly in order to determine which spaces \(X\) have \(C^*(X)\neq C(X)\), he named, defined, gave examples of non-countably compact, and studied some properties of \textit{pseudocompact} spaces, those Tychonoff spaces \(X\) for which \(C^*(X)=C(X)\). Then beginning in the early 1950s and continuing to the present, pseudocompact and related spaces have been studied by numerous researchers. Noting this, the editors of the book being reviewed (``the book'') remark that while a small number of well-known books on topology include a considerable amount of information on pseudocompactness, it is also the case that ``in spite of the importance of this concept, there is as yet no text that systematically compiles and develops the, by now, extensive theory of pseudocompact spaces.'' They go on to state that their goal is ``to correct, at least in part, this absence, to present in a book, intended for postgraduate students and researchers, many of the results of historical importance on this subject, and to develop material which has not been published in any previous work of this kind.'' The editors and authors have included an enormous amount of material in their well-written book, and the reviewer certainly feels they have achieved their goal. The book consists of eight chapters varying in length from 26 to 44 pages. Each chapter presents in an orderly way introductory statements, theorems, lemmas, most with proofs, questions, historical comments, and a reasonably up-to-date list of 31 to 86 references. Each chapter is designed to be mostly self-contained, so while working through proofs in one chapter the reader may need only occasionally to review some proof or statement in a different chapter or other publication. Some detailed…
reviewed by: Robert M. Stephenson Jr.
Mathematics Subject Classification ID: 54-06
Mathematics Subject Classification ID: 54D30
Mathematics Subject Classification ID: 54D20
Mathematics Subject Classification ID: 00B15
zbMATH DE Number: 6897708
zbMATH Keywords: pseudocompact
zbMATH Keywords: feebly compact
zbMATH Keywords: maximal pseudocompact
zbMATH Keywords: Mrówka-Isbell \(\Psi\)-space
zbMATH Keywords: pseudocompact group
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-319-91680-4
OpenAlex ID: W2984062968
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description: scientific article; zbMATH DE number 6897708