Proximity approach to problems in topology and analysis.

*(English)*Zbl 1185.54001
München: Oldenbourg Verlag (ISBN 978-3-486-58917-7/pbk). xiv, 204 p. (2009).

This book is an interesting summary of the author’s more than 45 years of research in general topology, which is grounded in the conviction that the concept of nearness of sets plays a major role in solutions to topological problems. Topological spaces are the result of the axiomatization of the idea of nearness of points to sets. In 1951, V. A. Efremovič introduced proximity spaces in [Dokl. Akad. Nauk SSSR, n. Ser. 76, 341–343 (1951; Zbl 0042.16703)] by axiomatizing nearness of sets to sets. It was the author’s book from 1970, written jointly with B. D. Warrack, that brought Efremovič’s proximity spaces to the attention of a wider audience [Proximity spaces. Cambridge Tracts in Mathematics and Mathematical Physics. No. 59. London: Cambridge University Press, X, 128 p. (1970; Zbl 0206.24601)]. Since then, he and many others have used proximities in various areas of general topology to solve problems and to generalize, unify, and simplify known results. Much of this work is exhibited in the present book, which consists of 15 chapters.

In Chapter 1, Lodato proximities are introduced, a generalization of Efremovič proximities. It is shown that for every compatible Lodato proximity \(\delta\) on a \(T_1\)-space \(X\) a \(T_1\)-compactification \(X^\delta\) of \(X\) can be constructed in a natural way. Since every \(T_1\)-compactification of a \(T_1\)-space can be obtained in this way, the proximity approach unifies many constructions that have been studied separately before. Chapter 2 is devoted to Efremovič proximities. If \(\delta\) is a compatible Efremovič proximity on a completely regular \(T_1\)-space \(X\), then \(X^\delta\) is a Hausdorff compactification of \(X\), called the Smirnov compactification. Again, every Hausdorff compactification of a completely regular \(T_1\)-space can be obtained in this way. In Chapter 3, Efremovič proximities are used to prove a generalization of the celebrated Taimanov extension theorem. Chapter 4 is a brief introduction to nearness spaces, which H. Herrlich introduced in [General Topology Appl. 4, 191–212 (1974; Zbl 0288.54004)] by axiomatizing the idea of nearness of a family of sets. It is shown how the generalized Taimanov extension theorem can further be extended to regular nearness spaces. Most of these results were proved between 1970 and 1980.

Chapters 5 to 8 contain more recent results. They are devoted to the investigation of topologies on the set \(\text{CL}(X)\) of nonempty closed subsets of a \(T_1\)-space \((X,\tau)\), i.e., to hyperspaces. Recall that the classical Vietoris topology on \(\text{CL}(X)\) can be described as follows. For each \({\mathcal U}\in[\tau]^{<\omega}\) let \({\mathcal U}^-\) be the subset of \(\text{CL}(X)\) consisting of all \(A\in\text{CL}(X)\) with \(A\cap U\neq\emptyset\) for each \(U\in{\mathcal U}\), of all \(A\in\text{CL}(X)\) that hit each \(U\in{\mathcal U}\). Then \(\tau(V^-)= \{{\mathcal U}^-\mid{\mathcal U}\in [\tau]^{<\omega}\}\) is a basis for a topology, the so-called lower Vietoris topology on \(\text{CL}(X)\). Moreover, for each \(U\in\tau\) let \(U^+\) be the subset of \(\text{CL}(X)\) consisting of all \(A\in\text{CL}(X)\) with \(A\subset U\), i.e., of all \(A\in\text{CL}(X)\) which miss the closed set \(X\setminus U\). Then \(\tau(V^+)= \{U^+\mid U\in\tau\}\) is a basis for a topology, the so-called upper Vietoris topology on \(\text{CL}(X)\). The Vietoris topology \(\tau(V)\) is the join of \(\tau(V^-)\) and \(\tau(V^+)\) on \(\text{CL}(X)\). For obvious reasons it is called a hit-and-miss topology. In Chapter 5 it is shown that virtually every hyperspace topology that has been studied in the literature can be viewed as a hit-and-miss-topology, in particular the Fell topology, the Hausdorff metric topology, and the Wijsman topology. This observation, first published in the author’s paper [Appl. Gen. Topol. 3, No. 1, 45–53 (2002; Zbl 1033.54006)], greatly simplifies the investigation of hyperspace topologies.

Generalizing the Wijsman topology using Lodato proximities, a new hyperspace topology, called Bombay topology, is introduced in Chapter 6. It turns out that nearly every hyperspace topology considered to date is a special Bombay topology. Moreover, by combining the hit and miss parts of known topologies, dozens of new hyperspace topologies are obtained. A comprehensive survey can be found in [G. Di Maio and S. A. Naimpally, Ric. Mat. 51, No. 1, 49–60 (2002; Zbl 1146.54305)]. Chapter 7 contains some results concerning the supremum and the infimum of hyperspace topologies induced by metrics. Hyperspace topologies give rise to function space topologies. Some of them are considered in Chapters 8 and 9. The remaining chapters are devoted to different applications of proximities. In Chapters 10 and 11, metrizability and developability are characterized in terms of proximities. In Chapter 12, open and uniformly open relations are studied, and Chapter 13 is concerned with duality in function spaces. Chapter 14 contains proximity characterizations of some uniform invariants, and the final Chapter 15 presents select applications.

Altogether, this is an interesting book. I enjoyed reading it. However, it is not organized as a usual textbook. Most of the chapters can be read independently of each other. Definitions are repeated whenever they are needed, sometimes – if necessary – several times. Proofs are usually short. In fact, it is more a collection of survey articles than a textbook. Overall, it is a fairly personal view of the work done and stimulated by the author.

In Chapter 1, Lodato proximities are introduced, a generalization of Efremovič proximities. It is shown that for every compatible Lodato proximity \(\delta\) on a \(T_1\)-space \(X\) a \(T_1\)-compactification \(X^\delta\) of \(X\) can be constructed in a natural way. Since every \(T_1\)-compactification of a \(T_1\)-space can be obtained in this way, the proximity approach unifies many constructions that have been studied separately before. Chapter 2 is devoted to Efremovič proximities. If \(\delta\) is a compatible Efremovič proximity on a completely regular \(T_1\)-space \(X\), then \(X^\delta\) is a Hausdorff compactification of \(X\), called the Smirnov compactification. Again, every Hausdorff compactification of a completely regular \(T_1\)-space can be obtained in this way. In Chapter 3, Efremovič proximities are used to prove a generalization of the celebrated Taimanov extension theorem. Chapter 4 is a brief introduction to nearness spaces, which H. Herrlich introduced in [General Topology Appl. 4, 191–212 (1974; Zbl 0288.54004)] by axiomatizing the idea of nearness of a family of sets. It is shown how the generalized Taimanov extension theorem can further be extended to regular nearness spaces. Most of these results were proved between 1970 and 1980.

Chapters 5 to 8 contain more recent results. They are devoted to the investigation of topologies on the set \(\text{CL}(X)\) of nonempty closed subsets of a \(T_1\)-space \((X,\tau)\), i.e., to hyperspaces. Recall that the classical Vietoris topology on \(\text{CL}(X)\) can be described as follows. For each \({\mathcal U}\in[\tau]^{<\omega}\) let \({\mathcal U}^-\) be the subset of \(\text{CL}(X)\) consisting of all \(A\in\text{CL}(X)\) with \(A\cap U\neq\emptyset\) for each \(U\in{\mathcal U}\), of all \(A\in\text{CL}(X)\) that hit each \(U\in{\mathcal U}\). Then \(\tau(V^-)= \{{\mathcal U}^-\mid{\mathcal U}\in [\tau]^{<\omega}\}\) is a basis for a topology, the so-called lower Vietoris topology on \(\text{CL}(X)\). Moreover, for each \(U\in\tau\) let \(U^+\) be the subset of \(\text{CL}(X)\) consisting of all \(A\in\text{CL}(X)\) with \(A\subset U\), i.e., of all \(A\in\text{CL}(X)\) which miss the closed set \(X\setminus U\). Then \(\tau(V^+)= \{U^+\mid U\in\tau\}\) is a basis for a topology, the so-called upper Vietoris topology on \(\text{CL}(X)\). The Vietoris topology \(\tau(V)\) is the join of \(\tau(V^-)\) and \(\tau(V^+)\) on \(\text{CL}(X)\). For obvious reasons it is called a hit-and-miss topology. In Chapter 5 it is shown that virtually every hyperspace topology that has been studied in the literature can be viewed as a hit-and-miss-topology, in particular the Fell topology, the Hausdorff metric topology, and the Wijsman topology. This observation, first published in the author’s paper [Appl. Gen. Topol. 3, No. 1, 45–53 (2002; Zbl 1033.54006)], greatly simplifies the investigation of hyperspace topologies.

Generalizing the Wijsman topology using Lodato proximities, a new hyperspace topology, called Bombay topology, is introduced in Chapter 6. It turns out that nearly every hyperspace topology considered to date is a special Bombay topology. Moreover, by combining the hit and miss parts of known topologies, dozens of new hyperspace topologies are obtained. A comprehensive survey can be found in [G. Di Maio and S. A. Naimpally, Ric. Mat. 51, No. 1, 49–60 (2002; Zbl 1146.54305)]. Chapter 7 contains some results concerning the supremum and the infimum of hyperspace topologies induced by metrics. Hyperspace topologies give rise to function space topologies. Some of them are considered in Chapters 8 and 9. The remaining chapters are devoted to different applications of proximities. In Chapters 10 and 11, metrizability and developability are characterized in terms of proximities. In Chapter 12, open and uniformly open relations are studied, and Chapter 13 is concerned with duality in function spaces. Chapter 14 contains proximity characterizations of some uniform invariants, and the final Chapter 15 presents select applications.

Altogether, this is an interesting book. I enjoyed reading it. However, it is not organized as a usual textbook. Most of the chapters can be read independently of each other. Definitions are repeated whenever they are needed, sometimes – if necessary – several times. Proofs are usually short. In fact, it is more a collection of survey articles than a textbook. Overall, it is a fairly personal view of the work done and stimulated by the author.

Reviewer: H. Brandenburg (Berlin)

##### MSC:

54-02 | Research exposition (monographs, survey articles) pertaining to general topology |

54E05 | Proximity structures and generalizations |

54B20 | Hyperspaces in general topology |

54D35 | Extensions of spaces (compactifications, supercompactifications, completions, etc.) |

54E30 | Moore spaces |

54E35 | Metric spaces, metrizability |