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.
