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**A \(C_p\)-theory problem book. Topological and function spaces.**
*(English)*
Zbl 1222.54002

Problem Books in Mathematics. Berlin: Springer (ISBN 978-1-4419-7441-9/hbk; 978-1-4419-7442-6/ebook). xvi, 485 p. (2011).

The term \(C_p\)-theory was coined by Arhangel’skii to mean the study of spaces of continuous functions with the topology of pointwise convergence. The author, with many contributions of his own to this field, has written a guidebook for the reader to reach the frontiers of this field as of 2010 without first having to master a large amount of general topology. The reader is assumed to have some background in set theory and the topology of the real line. The author first poses 500 problems in \(C_p\)-theory and then gives their solutions in detailed outline form.

There are 100 problems on each of the following basic notions of topology and function spaces: (i) topologies, separation axioms and a glance at \(C_p(X)\); (ii) products, cardinal functions and convergence; (iii) metrizability and completeness; (iv) compactness type properties in function spaces; (v) more on completeness: real-compact spaces. A detailed summary of exercises divides each basic notion into a number of topics. Each topic is introduced with some definitions and basic facts. Some wider implications for general topology from these solutions are then set down under the heading bonus results.

Finally, there is a list of open problems with bibliographical citations and lively comments by the author. The Bibliography and Index are adequate for a reader working his way through the book but not as a reference source for other mathematicians. The author’s hope is to make \(C_p\)-theory accessible to talented undergraduates. His intention is that the reader can solve any given problem in the book if he has understood the solutions of a good proportion of all previous problems. He offers anecdotal evidence to show that this learning method does get results. It lies near the other end of the spectrum from the method of R. L. Moore. This book is a welcome addition to the field of general topology and \(C_p\)-theory in particular.

There are 100 problems on each of the following basic notions of topology and function spaces: (i) topologies, separation axioms and a glance at \(C_p(X)\); (ii) products, cardinal functions and convergence; (iii) metrizability and completeness; (iv) compactness type properties in function spaces; (v) more on completeness: real-compact spaces. A detailed summary of exercises divides each basic notion into a number of topics. Each topic is introduced with some definitions and basic facts. Some wider implications for general topology from these solutions are then set down under the heading bonus results.

Finally, there is a list of open problems with bibliographical citations and lively comments by the author. The Bibliography and Index are adequate for a reader working his way through the book but not as a reference source for other mathematicians. The author’s hope is to make \(C_p\)-theory accessible to talented undergraduates. His intention is that the reader can solve any given problem in the book if he has understood the solutions of a good proportion of all previous problems. He offers anecdotal evidence to show that this learning method does get results. It lies near the other end of the spectrum from the method of R. L. Moore. This book is a welcome addition to the field of general topology and \(C_p\)-theory in particular.

Reviewer: James V. Whittaker (Vancouver)

### MSC:

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

00A07 | Problem books |

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

54C35 | Function spaces in general topology |