# A Cp-Theory Problem Book

## Functional Equivalencies

Part of the Problem Books in Mathematics book series (PBM)

Part of the Problem Books in Mathematics book series (PBM)

This fourth volume in Vladimir Tkachuk's series on *Cp*-theory gives reasonably complete coverage of the theory of functional equivalencies through 500 carefully selected problems and exercises. By systematically introducing each of the major topics of *Cp*-theory, the book is intended to bring a dedicated reader from basic topological principles to the frontiers of modern research. The book presents complete and up-to-date information on the preservation of topological properties by homeomorphisms of function spaces. An exhaustive theory of *t*-equivalent, *u*-equivalent and *l*-equivalent spaces is developed from scratch. The reader will also find introductions to the theory of uniform spaces, the theory of locally convex spaces, as well as the theory of inverse systems and dimension theory. Moreover, the inclusion of Kolmogorov's solution of Hilbert's Problem 13 is included as it is needed for the presentation of the theory of *l*-equivalent spaces. This volume contains the most important classical results on functional equivalencies, in particular, Gul'ko and Khmyleva's example of non-preservation of compactness by *t*-equivalence, Okunev's method of constructing *l*-equivalent spaces and the theorem of Marciszewski and Pelant on *u*-invariance of absolute Borel sets.

*Cp*-theory and general topology as well as a reference guide for specialists working in *Cp*-theory and related topics. Additionally, the material can also be considered as an introduction to advanced set theory and descriptive set theory. …I hope that this book becomes the right hand of students and researchers in mathematics.” (**Mathematical Reviews**)

From the reviews of *Special Features of Function Spaces*: “This is a well-written and very interesting book from several points of view and can be used as a textbook for

courses in both From the reviews of *Topological and Function Spaces*: “It is designed to bring a dedicated reader from basic topological principles to the frontiers of modern research. The only background needed is some knowledge of set theory and real numbers. …This volume can also be used as a reference for mathematicians working in or outside of the field of topology (functional analysis) wanting to use results or methods of Cp-theory.” (**Mathematical Reviews**)

1-equivalence Cp-theory Lindelof Sigma-property Lindelof space additional axioms of ZFC cardinal functions compact space convergence properties function spaces metrizability open questions topology pointwise convergence set t heory t-equivalence u-equivalence homeomorphisms of function spaces linear topological spaces l-equivalence continuous maps Cp-spaces

- DOI https://doi.org/10.1007/978-3-319-24385-6
- Copyright Information Springer International Publishing Switzerland 2016
- Publisher Name Springer, Cham
- eBook Packages Mathematics and Statistics
- Print ISBN 978-3-319-24383-2
- Online ISBN 978-3-319-24385-6
- Series Print ISSN 0941-3502
- Series Online ISSN 2197-8506
- About this book