# Special Topics in Mathematics for Computer Scientists

## Sets, Categories, Topologies and Measures

• Ernst-Erich Doberkat
Textbook

1. Front Matter
Pages i-xx
2. Ernst-Erich Doberkat
Pages 1-108
3. Ernst-Erich Doberkat
Pages 109-279
4. Ernst-Erich Doberkat
Pages 281-425
5. Ernst-Erich Doberkat
Pages 427-683
6. Back Matter
Pages 685-719

### Introduction

This textbook addresses the mathematical description of sets, categories, topologies and measures, as part of the basis for advanced areas in theoretical computer science like semantics, programming languages, probabilistic process algebras, modal and dynamic logics and Markov transition systems.

Using motivations, rigorous definitions, proofs and various examples, the author systematically introduces the Axiom of Choice, explains Banach-Mazur games and the Axiom of Determinacy, discusses the basic constructions of sets and the interplay of coalgebras and Kripke models for modal logics with an emphasis on Kleisli categories, monads and probabilistic systems. The text further shows various ways of defining topologies, building on selected topics like uniform spaces, Gödel’s Completeness Theorem and topological systems. Finally, measurability, general integration, Borel sets and measures on Polish spaces, as well as the coalgebraic side of Markov transition kernels along with applications to probabilistic interpretations of modal logics are presented. Special emphasis is given to the integration of (co-)algebraic and measure-theoretic structures, a fairly new and exciting field, which is demonstrated through the interpretation of game logics.

Readers familiar with basic mathematical structures like groups, Boolean algebras and elementary calculus including mathematical induction will discover a wealth of useful research tools. Throughout the book, exercises offer additional information, and case studies give examples of how the techniques can be applied in diverse areas of theoretical computer science and logics. References to the relevant mathematical literature enable the reader to find the original works and classical treatises, while the bibliographic notes at the end of each chapter provide further insights and discussions of alternative approaches.

### Keywords

coalgebras over general and probabilistic spaces examples for the use of the Axiom of Choice measurable spaces and Markov transition systems proof techniques in games and measure theory proof techniques in topology and categories topologies for use in computer science

#### Authors and affiliations

• Ernst-Erich Doberkat
• 1
1. 1.Lewackerstr. 6 bMath ++ SoftwareBochumGermany

### Bibliographic information

• DOI https://doi.org/10.1007/978-3-319-22750-4
• Copyright Information Springer International Publishing Switzerland 2015
• Publisher Name Springer, Cham
• eBook Packages Computer Science
• Print ISBN 978-3-319-22749-8
• Online ISBN 978-3-319-22750-4
• Buy this book on publisher's site