Hausdorff gaps and limits
type: Item
instance of: scholarly article
title: Hausdorff gaps and limits
zbMATH Open document ID: 0821.54001
author: R. Frankiewicz
author: Paweł Zbierski
published in: Studies in Logic and the Foundations of Mathematics
publication date: 1995-06-14T00:00:00Z
review text: Many set-theoretic topologists have long felt that there is a real need for a book on \(\beta N\) since there has been so much work on the subject in the last twenty or more years. For this reason a book on \(\beta N\) would have to be quite long and difficult to write to do justice to the title. The book under review is not a book on \(\beta N\): ``Hausdorff gaps and limits'' is perhaps a good description of the focus. A limit is a maximal descending mod finite sequence of infinite subsets of \(\omega\) and a gap is essentially a chain in \([\omega]^ \omega\) (ordered mod finite) which has order type equal to \(\kappa \lambda^*\) where \(\kappa\) and \(\lambda\) are cardinals and \(\lambda^*\) is \(\lambda\) with the reverse order. These concepts arise naturally in many areas of mathematics. The focus is largely restricted, quite frankly, to the interests and contributions of the authors. There are 48 references, none to Frolík, two to Shelah and 12 to one of the authors. However the book is still quite substantial and worth reading. The review may seem to come down on the critical side but this is because the material chosen has a beauty which is its own best defense and it is on this basis that the book is a welcome addition to my library shelves. Some may have other ideas of what should be in a book about \(\beta N\) but if 200 pages are to be written about \(\beta N\) there is nothing wrong with the choices that the authors have made (leaning towards the more topological facets). The first four chapters provide a reasonable (necessarily quite focussed) introduction to the aspects of Boolean algebras most pertinent to topology via Stone duality, Hausdorff gaps, Parovichenko's CH characterization of \(\omega^*\), the space of uniform ultrafilters, and extensions of real-valued continuous functions especially with respect to \(F\)-spaces and extremally disconnected spaces. Chapter 11 is also a nice (and quick) introduction to forcing. I couldn't help but think in …
reviewed by: Alan Dow
Mathematics Subject Classification ID: 54-02
Mathematics Subject Classification ID: 03-02
Mathematics Subject Classification ID: 54D35
Mathematics Subject Classification ID: 03E10
Mathematics Subject Classification ID: 03E40
Mathematics Subject Classification ID: 54D40
Mathematics Subject Classification ID: 54D80
Mathematics Subject Classification ID: 03E50
zbMATH DE Number: 764213
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 764213