Hausdorff gaps and limits
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 terms of how this book might function as a graduate text as I read it. I'm reasonably pleased with its potential in this regard. Perhaps more work should have gone into the selection and organization of the problems (for example problems on extremally disconnected spaces are sprinkled randomly throughout) and the book is a bit uneven in its prerequisites (it uses \(\diamondsuit\) on demand but includes proofs that the cub filter is \(\text{cf} (\kappa)\)-complete and a whole section on stationary sets). While I'm still satisfied with its potential as a graduate text there is the very unfortunate fact that the table of contents is only accurate to within 10 pages. The main chapters (5 through 10) are essentially each devoted to a particular theorem. Chapter 5 introduces the base matrix of Balcar, Pelant, and Simon and then reproduces the proof that every ultrafilter on \(\omega\) has an almost disjoint refinement. Chapter 6 gives Kunen's construction of weak \(P\)-points and their application to inhomogeneity of \(F\)-spaces and products. In Chapter 7, discrete \(\omega_ 1\)-sequences in \(\omega^*\) are discussed. The first result is that there is a converging \(\omega_ 1\)-sequence; this result and those of Chapters 5 and 6 are noteworthy (not just for their beauty but also) for the fact that they are ZFC results. Chapter 7 finishes with Shelah's result that it is consistent that every strongly discrete \(\omega_ 1\)-sequence is actually \(C^*\)-embedded (the opposite, in a sense, of convergent). Chapter 8, ``The Martin's Axiom'', is probably mistitled. The main results here are to show that various statements are consistent with or independent of Martin's Axiom. These include the existence of \((\omega_ 1, {\mathfrak c})\)-gaps, Parovichenko's theorem (but not with full MA), and the automatic continuity of linear homomorphisms on Banach spaces. This is not a good reference work because of poor detail in historical facts and references. It tends to refer to the most recent appearances of results (maybe with minor improvements on the originals) rather than to celebrate the important work of earlier contributors. The authors do not provide any guidance for future work. There is no list of open problems and no real philosophy expressed. While there are typos they do not interfere with the readability which is generally quite good. The proofs are set at a level that a good graduate student could follow. There are likely some places that the forcing prerequisites become too extreme but on the whole some experience with Martin's Axiom should suffice as background (but one will have to read the Forcing section in the book and be prepared to learn on the fly). As indicated above, this book would have benefitted from more attention being paid to the references. Many of the problems are excerpts from published papers. Mentioning these in some historical notes would have been very helpful to anyone interested in tracking down original citations. Some other citations that probably should have been included are listed next. 4.3.7 is used (but not for the first time) in van Mill's 1984 article in the Handbook of Set-Theoretic topology, Theorem 1.4.4. Shelah was the first to prove 4.6.1, it appears in his first Classification Theory book (there are ultrafilters on \(\omega\) so that the ultrapower has \((\omega, \omega_ 1)\)-gaps). It is also easily deduced from 7.2.3 (Balcar et. al.'s long converging \(\omega_ 1\)- sequence construction -- get the limit point to also be a limit point of a countable discrete subset of its fiber). 4.6.2 is due to the reviewer (1984 Topology Proceedings) and the reference [FZ\(_ 1\)]should be 1987 not 1978. 6.3.5 and 6.4.2 are due to Kunen (1978 widely circulated preprint). 7.3.1 is due to Shelah (1978 widely circulated manuscript ``Some consistency results in topology''). Some of the problems have slight errors in them. The ones that I noticed are listed here. 3.E.1 \(Y\) needs to be dense or perhaps it is meant that \(\overline Y\) is homeomorphic to \(\beta [Y]\). In 3.F.2 the assumptions on \(D\) are not sufficient to guarantee that \(\overline D\) is homeomorphic to \(\beta [D]\). In 4.G.2 I believe \(A\) must be a zero set of \(\beta [X]\) (this may not be an error but it isn't clear what is meant). Problem 5.H.2 asks us to show that every chain in \({\mathcal B} (\kappa 1)\) of length \(\kappa\) generates the whole of \({\mathcal B} (\kappa 1)\); maybe generates an algebra which is isomorphic to \({\mathcal B} (\kappa 1)\) was intended. 8.A.1 (c) \(\mathbb{S}_ \kappa\) is an iteration of \(\mathbb{S} (R_ \alpha)\) where the \(R_ \alpha\) are growing (appropriately). Finally here are some other general comments. Regular open sets are referred to as open domains. In my opinion \(\S3\) of Chapter 5 should have tried harder to illustrate the important role that \(\pi\)-base matrices play in this important theorem since it is a very useful technique. It is an unfortunate omission to not mention in 6.3.5 that Simon has proven that there are \(2^{2^{\mathfrak c}}\) RK-incomparable weak \(P\)-points. It would have been natural to mention that M. Rabus has shown that 7.3.1 is independent of MA plus \({\mathfrak c} = \omega_ 2\). It is very disappointing that the full MA proof in 8.3.1 is omitted (and to my knowledge has not appeared elsewhere) but the present proof is essentially repeated again for Theorem 4.1 of Chapter 9. Problem 8.F.3 outlines a proof that ``if \(\omega < \kappa < {\mathfrak c}\) then \(\omega^* \not \equiv \kappa^*\)'' follows from MA but there is a much more direct proof: \(\{\kappa - \alpha : \alpha < \kappa\}\) is a ``limit'' (strange term) in \({\mathcal P} (\kappa)/fin\) but \({\mathcal P} (\omega)/fin\) has already been shown to not have such limits. The generalized \(\Delta\)-system lemma and the topological dynamics seem quite out of place in the Chapter 9 problem list.
scientific article; zbMATH DE number 764213