Pseudocompact topological spaces. A survey of classic and new results with open problems
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 comments about the topics studied are the following, where \textit{all hypothesized spaces are Tychonoff}, unless indicated otherwise. Chapter 1 (``Basic and classic results on pseudocompact spaces'', by \textit{J. Angoa-Amador}, \textit{A. Contreras-Carreto}, \textit{M. Ibarra-Contreras} and \textit{Á. Tamariz-Mascarúa}) contains: a number of Hewitt's and other authors' characterizations of pseudocompactness, such as the property, every locally finite family of open sets is finite (which for arbitrary topological spaces is usually called \textit{feebly compact}); proofs that pseudocompactness is preserved by regular closed sets, continuous maps, and extensions; examples such as the Tychonoff plank and an Isbell-Mrówka \(\Psi\)-space; discussion of the relationships among pseudocompact, countably compact, Baire, and metacompact spaces; some product space theorems, such as (a) every product of pseudocompact \(k\)-spaces is pseudocompact, and (b) I.\ Glicksberg's correction of Hewitt's incorrect assertion (that the Stone-Čech compactification of a product is the product of Stone-Čech compactifications of the factor spaces) obtained by assuming the product space is pseudocompact; and when \(C_p(X,K)\), where \(K\) is compact metrizable, is pseudocompact. In a ``Final Note,'' the authors provide references containing further important results such as suitable conditions on a product space or on a mapping that pseudocompactness be preserved or be an inverse invariant of the mapping. Chapter 2 (``Pseudocompact topological groups'', by \textit{M. Tkachenko}) includes a very good introduction to topological groups and uniformities on them and develops in detail properties of pseudocompact groups, subgroups of pseudocompact groups, precompact groups, feebly compact paratopological groups (groups in which multiplication is assumed to be jointly continuous, but inversion need not be continuous, and the topology may not even be regular), and semiregularizations of paratopological groups, and derives the Comfort-Ross theorem that the property, pseudocompact group, is productive. In addition, a proof is given that the property, feebly compact paratopological group, is also productive. Chapter 3 (``Pseudocompactness and ultrafilters'', by \textit{S. García-Ferreira} and \textit{Y. F. Ortiz-Castillo}) begins with an introductory discussion and provides a long list of characterizations of pseudocompactness which extends the corresponding list in Chapter 1. Then it proceeds to a study of \(p\)-pseudocompactness and similar types of pseudocompactness, all of which imply pseudocompact\-ness, and some of which are stronger than \(p\)-pseudocompactness, and others of which are weaker than \(p\)-pseudocompact\-ness. The notion of \(p\)-pseudocompactness was first defined and studied in a 1975 article by J.\ Ginsberg and V.\ Saks, who modified A.R.\ Bernstein's notion of \(p\)-compact spaces. In particular, for a free ultrafilter \(p\) on \(\mathbb N\), they defined a space \(X\) to be \(p\)-\textit{pseudocompact} if for every sequence \(\{U_n\}\) of nonempty open subsets of \(X\), there is a point \(x\in X\) such that for every neighborhood \(V\) of \(x\), \(\{n:U_n\cap V\neq\emptyset\}\in p\). Ginsberg and Saks proved that \(p\)-pseudocompactness is productive and implies pseudocompactness, and they derived some of its other properties. They also studied densely \(p\)-pseudocompact spaces and gave a number of examples, one of which is the space \(X=\Pi\{\beta{\mathbb N}\setminus \{x\}:x\in\beta{\mathbb N} \setminus{\mathbb N}\} \), which is not \(p\)-pseudocompact for any \(p\in{\mathbb N}^*\), but is such that every power of \(X\) is pseudocompact. (Although the authors include the 1975 Ginsberg and Saks article in the references of Chapter 3 as item [\textit{J. Ginsburg}, Can. J. Math. 27, 1392--1399 (1975; Zbl 0314.54007)] and attribute one theorem to it, they state that the definition of a \(p\)-limit of a sequence of sets was ``introduced in [\textit{S. García-Ferreira} and \textit{M. Sanchis}, Houston J. Math. 23, No. 1, 65--86 (1997; Zbl 0881.54019)]'' and note only that the above product example appeared in [loc. cit.]; the author of Chapter 4 correctly attributes the preceding definition to Ginsberg and Saks.) After a review of the property \(p\)-pseudocompact, some of the specific concepts the authors of Chapter 3 study are \((\alpha,D)\)-pseudocompact, ultrapseudocompact, pseudo-\(\omega\)-bounded, strongly \(p\)-pseudocompact, and strongly pseudocompact. They present an extensive number of results concerning the very nice behavior of spaces having these properties, and include such topics as groups, hyperspaces, types of points, and the Rudin-Keisler pre-order on \({\mathbb N}^*\). The title of Chapter 4 is ``Bounded subsets of Tychonoff spaces: a survey of results and problems'' (by \textit{M. Sanchis}). A subset \(B\) of a space \(X\) is said to be \textit{bounded in \(X\)} provided that every function \(f\in C(X)\) is bounded on \(B\). The first two sections of this chapter provide a well-written historical and bibliographical discussion of the occurrence of this notion in the literature and then develop characterizations and basic properties of bounded subsets of spaces. Examples of the characterization results presented are the equivalence of the following: \(B\) is bounded in \(X\); every countable locally finite family of pairwise disjoint open sets of \(X\), each member of which meets \(B\), is finite; the closure of \(B\) is the same in each of the spaces \(\mu X\) (the Dieudonné completion of \(X\)), \(\upsilon X\) (the Hewitt realcompactification of \(X\)), and \(\beta X\) (the Stone-Čech compactification of \(X\)); and for every compatible uniformity \(\mathcal U\) on \(X\), \(B\) is precompact in \((X,\mathcal U)\). Two other properties noted are: the closure of an open bounded subset is pseudocompact; and a closed bounded subset of a normal space is countably compact. In the remaining sections, some variations on boundedness are investigated which lead to modifications of theorems concerning pseudocompactness by such authors as W. W. Comfort and A. K. Ross, Z. Frolík, Ginsberg and Saks, and Glicksberg. For example, a subset \(S\) of a space \(X\) is called \textit{strongly bounded in \(X\)} if for each space \(Y\) and bounded subset \(B\) of \(Y\), the set \(S\times B\) is bounded in \(X\times Y\). A characterization due to Frolík is generalized to characterize strongly bounded subsets. For another, a subset \(B\) is said to be \(p\)-\textit{bounded in a space \(X\)}, where \(p\) is a free ultrafilter on \(\mathbb N\), provided that every sequence \(\{U_n\}\) of nonempty open subsets of \(X\) with \(B\cap U_n\neq\emptyset\), for all \(n\), has a \(p\)-limit in \(X\). Properties of strongly bounded subsets and \(p\)-bounded subsets are derived. In the final section, the focus is on topological groups and their bounded subsets (equivalently, \(p\)-bounded subsets for some \(p\)), and a number of product theorems are presented. \par Chapter 5 (``Weakly pseudocompact spaces'', by \textit{A. Dorantes-Aldama}, \textit{O. Okunev} and \textit{Á. Tamariz-Mascarúa}) primarily deals with a weakening of the following characterization of pseudocompactness of a space \(Y\) given by Hewitt in his [loc. cit.] article: ``\(\beta Y\setminus Y\) contains no closed \(G_\delta\) except the void set.'' A space \(X\) is defined to be \textit{weakly pseudocompact} provided that it is \(G_\delta\)-dense in some compactification \(K\) of \(X\), i.e., provided every nonempty \(G_\delta\)-subset of \(K\) has nonempty intersection with \(X\). Part of the interest in weakly pseudocompact spaces is due to properties they share with pseudocompact spaces and also nice properties they have which pseudocompact spaces do not have. Some of the results the authors present are these. Every weakly pseudocompact space is Baire. Every product of weakly pseudocompact spaces is weakly pseudocompact. A space is weakly pseudocompact if and only if it is \(G_\delta\)-dense in a locally pseudocompact space. Zero-sets of pseudocompact spaces are weakly pseudocompact, and Lindelöf weakly pseudocompact spaces are compact. Results are discussed concerning weak pseudocompactness and GO-spaces, topological groups, \(C_p(X,Y)\), and other types of spaces, and examples and theorems are given concerning whether or not, and under what conditions, weak pseudocompactness is preserved by unions, continuous images, subspaces, and factor spaces. As discussed in its introductory section, the primary focus of Chapter 6 (``Maximal pseudocompact spaces'', by \textit{M. Madriz-Mendoza}, \textit{V. V. Tkachuk} and \textit{R. G. Wilson}) is presented in Sections 6.3--6.6 on maximal properties in spaces in which the word ``space'' has the same meaning as it does in most of the book. Then in Section 6.2, and only in that section, the word ``space'' may refer to a topological space which is not necessarily Tychonoff, and saying a space is ``pseudocompact'' just means \(C(X)=C^*(X)\). The reader will notice that 6.2 mainly states some pre-1995 results and outlines of proofs which are not too involved, and 6.3--6.6 provide detailed, mostly post-2010 results and proofs. In this review, the meaning of the expression ``maximal \(\mathcal P\)'' will have its older meaning when we refer to 6.2, and it will abbreviate ``maximal \(\mathcal P\) in the class of Tychonoff spaces'' when we refer to later sections of the chapter (a different convention is used in the book). Here are some results from 6.2: A pseudocompact space \((X,\tau)\) is maximal pseudocompact if and only if for every subset \(F\) of \(X\), if \(F\) is bounded in \((X,\sigma)\), where \(\sigma\) is the topology generated by \(\tau\cup\{X\setminus F\}\), then \(F\) is closed in \((X,\tau)\). A feebly compact space \(X\) is maximal feebly compact if and only if \(X\) is submaximal (every dense set is open) and every feebly compact subspace of \(X\) is closed. A regular feebly compact space is maximal feebly compact if and only if it is homeomorphic with \(\Psi(S,\mathcal A)\) (defined later in this review) for some MAD family \(\mathcal A\) on a discrete space \(S\). (While feebly compact subspaces of any such \(\Psi(S,\mathcal A)\) are closed subsets, the authors' proof outline of that fact and their statement that the space is first countable need to be modified since members of \(\mathcal A\) may be uncountable.) In 6.3--6.6 the authors obtain characterization, product, continuous image, and subspace theorems, provide a number of examples, and present some unsolved problems. A few of the theorems obtained: A pseudocompact space \((X,\tau)\) is maximal pseudocompact if and only if (i) each non-isolated point is the unique accumulation point of some countable family of open sets, and (ii) whenever \(\mathcal A\) is a family of non-open sets, each of which is contained in the closure of its interior, and \(\mathcal A\) is such that the topology generated by \(\tau\cup\mathcal A\) is Tychonoff, then that topology is not pseudocompact. A pseudocompact space with a dense set of isolated points \(D\) and which is strongly accessible from \(D\) is maximal pseudocompact (and maximal regular feebly compact). A dyadic space (paratopological group) is maximal pseudocompact if and only if it is metrizable (and compact). A linearly ordered space is maximal pseudocompact if and only if it is countably compact and first countable. The product of a maximal pseudocompact space and a countable compact space is maximal pseudocompact. The product of two maximal pseudocompact spaces need not be maximal pseudocompact. One topic which might have been presented briefly in Section 6.2 or another chapter of the book is characterizing \(\mathcal P\)-minimal spaces and \(\mathcal P\)-closed spaces, for spaces not necessarily Tychonoff, for one of the compound properties \(\mathcal P\) equal to: Hausdorff (regular \(T_1\), Tychonoff, zero-dimensional \(T_1\)) and first countable; semimetrizable and Hausdorff; or Moore. Proofs published in [\textit{R. M. Stephenson jun.}, Topology Appl. 27, 11--28 (1987; Zbl 0633.54012)] or in one of the references of that article showed that for any such \(\mathcal P\), a \(\mathcal P\)-space is \(\mathcal P\)-closed if and only if it is feebly compact, and a (semiregular) \(\mathcal P\)-space is \(\mathcal P\)-minimal if and only if it is feebly compact. Chapter 7 (``Pseudocompactness in the realm of topological transformation groups'', by \textit{N. Antonyan}, \textit{S. Antonyan} and \textit{M. Sanchis}) provides a well-organized review of pseudocompactness and compactifications for \(G\)-spaces. Given a Hausdorff topological group \(G\), a \textit{topological transformation group} or a \textit{\(G\)-space} with \textit{phase group \(G\)} is a triple \(\langle X,G,\alpha\rangle\), where \(X\) is a topological space and \(\alpha:G\times X\rightarrow X\) is a continuous map such that for all \(g_1, g_2\in G\) and \(x\in X\), \(\alpha(e,x)=x\) and \(\alpha(g_1,\alpha (g_2,x))=\alpha(g_1g_2,x)\). A \(G\)-space with \(G=\mathbb R\) (\(G=\mathbb Z\)) is called a \textit{continuous} (\textit{discrete}) \textit{flow}, and the authors note that continuous and discrete flows are the main objects of study in dynamical systems. They study characterizing \(G\)-psudocompact spaces and the equalities such as \(\beta_G X=\beta X\) (where \(\beta_G X\) denotes the maximal \(G\)-compactification of \(X\) with respect to a certain relation \(\preceq\)) and \(\beta_G(X\times Y)=\beta_G X\times\beta_G Y\). A \(G\)-space is called \(G\)-\textit{pseudocompact} if every \(G\)-uniform function \(f:X\rightarrow\mathbb R\) is bounded. One theorem developed is: For any group \(G\) and \(G\)-Tychonoff spaces \(X\) and \(Y\), if \(X\times Y\) is pseudocompact, then \(\beta_G(X\times Y)=\beta_G\times\beta_G Y\). Another is proving there is a locally compact countably compact \(G\)-Tychonoff space \(Y\) such that \(\beta_G Y=\alpha Y\neq\beta Y\). Chapter 8 (``Topology of Mrówka-Isbell \(\Psi\)-spaces'' by \textit{F. Hernández-Hernández} and \textit{M. Hrušák}) is a wide-ranging survey. We recall that a family \(\mathcal A\) of infinite subsets of an infinite set \(S\) is said to be \textit{almost disjoint} (\(AD\)) if the intersection of any two distinct members of \(\mathcal A\) is finite, and an \(AD\) family \(\mathcal A\) of infinite subsets of \(S\) is said to be \textit{maximal almost disjoint} (\(MAD\)) if \(\mathcal A\) is contained in no other \(AD\) family of infinite subsets of \(S\). In either case, \(\Psi(S,\mathcal A)\) denotes the set \(S\cup \mathcal A\), topologized so that each point of \(S\) is isolated, and a neighborhood of a set \(A\in\mathcal A\) is any set of the form \(\{A\}\cup(A\setminus F)\), where \(F\) is any finite subset of \(S\). P.~Alexandroff and P.~Urysohn in 1926 described such a space and its properties, where they used the rationals and irrationals and an \(AD\) family, and then years later, S.~Mrówka and J.~Isbell independently studied spaces of the form \(\Psi(\omega,\mathcal A)\), where \(\omega\) is the first infinite ordinal, and \(\mathcal A\) is an infinite \(MAD\) family on \(\omega\). Spaces \(\Psi(S,\mathcal A)\) are defined and studied briefly in Chapter 6. The authors of Chapters 1 and 8 use the phrase \textit{Mrówka-Isbell space}, or more briefly, \(\Psi\)-\textit{space}, and write ``\(\Psi(\mathcal A)\)'' to refer to any \(\Psi(\omega,\mathcal A)\)-space in which \(\mathcal A\) is required to be an \textit{infinite} \(AD\) family on \(\omega\). The authors of Chapter 8 provide a large amount of information about the properties and applications of \(\Psi\)-spaces, with primary emphasis on ones in which \(\mathcal A\) is a \(MAD\) family. Some of the properties derived and theorems established or stated are the following. Each \(\Psi(\mathcal A)\) is a first countable, separable, locally compact Hausdorff, zero-dimensional Moore space which is not countably compact, and furthermore, \(\Psi(\mathcal A)\) is pseudocompact if and only if \(\mathcal A\) is \(MAD\). There is a \(MAD\) family \(\mathcal A\) on \(\omega\) such that \(|\mathcal A|=\mathfrak c\). There is a separable normal non-metrizable Moore space if and only if there is an uncountable normal \(\Psi\)-space. Letting \(\mathcal F(\mathcal A)\) denote the one-point compactification of \(\Psi(\mathcal A)\) for an \(AD\) family \(\mathcal A\), there is a \(MAD\) family \(\mathcal A\) such that \(\mathcal F(\mathcal A)=\beta(\Psi(\mathcal A))\). There is a \(MAD\) family \(\mathcal A=\mathcal A_0\cup\mathcal A_1\) such that \(\mathcal F(\mathcal A_i)\) is a Fréchet space for \(i\in 2\), but \(\mathcal F(\mathcal A_0)\times\mathcal F(\mathcal A_1)\) is not. If \(X\) is a compact metric space without isolated points, or if \(X\) is the ordinal \(\gamma+1\) with the order topology, for an ordinal \(\gamma<\mathfrak t^+\), then there is a \(MAD\) family \(\mathcal A\) such that \(\beta(\Psi(\mathcal A))\setminus\Psi(\mathcal A)\) is homeomorphic with \(X\). A few of the other topics studied in this chapter are completely separable \(MAD\) families, hyperspaces and products of \(\Psi\)-spaces, and spaces of continuous functions on \(\Psi\)-spaces.
scientific article; zbMATH DE number 6897708