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**A \(C_p\)-theory problem book. Compactness in function spaces.**
*(English)*
Zbl 1325.54001

Problem Books in Mathematics. Cham: Springer (ISBN 978-3-319-16091-7/hbk; 978-3-319-16092-4/ebook). xv, 524 p. (2015).

This is the third volume of author’s series [V. V. Tkachuk, A \(C_p\)-theory problem book. Topological and function spaces. Berlin: Springer (2011; Zbl 1222.54002); volume 2: Special features of function spaces. Problem Books in Mathematics. Cham: Springer (2014; Zbl 1303.54006)], devoted to compactness and compact-like properties in function spaces. The volume is divided into four chapters: 1. Behavior of compactness in function spaces (which contains 500 well selected problems, five sections with 100 problems each), 2. Solutions of problems 001–500 (complete solutions of problems formulated in Chapter 1), 3. Bonus results: some hidden statements (100 statements, called facts, which were used in the solutions of problems from Chapter 2; these statements are presented in 8 sections), 4. Open problems (100 open problems divided into 8 sections). It ends by 400 bibliographic items relevant in this field. The central part of Chapter 1 (and corresponding solutions in Chapter 2) concerns fundamental, up-to-date results about Corson, Gul’ko and Eberlein compacta and is presented in Sections 1.2–1.4. This is a good link with functional analysis. The statements in Chapter 3, called facts, may be considered as a kind of complementary material, and some of these facts are deep results and famous theorems (supplied with detailed proofs).

This volume, as the previous two, is a very useful book for all researchers working in \(C_p\)-theory (also in general topology) and its relationships with other mathematical disciplines, especially with functional analysis. The problems in Chapter 4 can attract young mathematicians to work in this field and to solve some of quite difficult problems.

This volume, as the previous two, is a very useful book for all researchers working in \(C_p\)-theory (also in general topology) and its relationships with other mathematical disciplines, especially with functional analysis. The problems in Chapter 4 can attract young mathematicians to work in this field and to solve some of quite difficult problems.

Reviewer: Ljubiša D. Kočinac (Niš)

### MSC:

54-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to general topology |

54C35 | Function spaces in general topology |

54D30 | Compactness |

46E99 | Linear function spaces and their duals |

00A07 | Problem books |

97U40 | Problem books, competitions, examinations (aspects of mathematics education) |