"The book...presents the contemporary theory of categorical closure operators, one of the main branches of categorical topology. The purpose of the theory is to develop a categorical characterization of the classical topological concepts. It provides a tool that allows us to extend these concepts to an arbitrary category so that the results of the theory have many applications in other fields of mathematics, particularly topology, algebra, and discrete mathematics.... The benefit of the book is that, after introducing the reader to the general theory, it presents a number of applications based on quite recent results achieved in this field. Thus the book provides basic literature for those interested in the theory of closure operators and its applications. It is an excellent exposition of a part of modern mathematics." — INTERNATIONALE MATHEMATISCHE NACHRICHTEN "The book under review puts its emphasis on giving a fairly elementary introduction in to the basic categorical theory of closure operators (Part I) and of their use in categorical topology (Part II), paying particular attention to the author's work on notions of connectedness... Each of the 17 section sof the book ends with a list of references for suggested further reading and a set of exercises, many of which ask the reader to verify claims made earlier in the text." ---ZENTRALBLATT MATH
Product Description
This book presents the general theory of categorical closure operators together with examples and applications to the most common categories, such as topological spaces, fuzzy topological spaces, groups, abelian and topological groups. The main aim of the theory, whose origin dates back to the 1980s, is to develop a categorical characterization of the classical basic concepts in topology via the newly introduced concept of categorical closure operator. This permits many topological ideas to be introduced in a topology-free environment and imported afterwards into a new category, which often yields interesting new insights into their structures.
The first part of the book deals with the general theory, starting with basic definitions and gradually moving to more advanced properties. The second part includes applications to the classical concepts of epimorphisms, separation, compactness and connectedness. Every chapter ends with exercises. A comprehensive list of references for the reader who wants to consult original works, and a good index complete the book.
Categorical Closure Operators is self-contained and can be considered as a graduate level text for topics courses in algebra, topology or category theory. The book appeals mainly to graduate students and researchers in categorical topology, and to those interested in categorical methods applied to the most common concrete categories. The reader is expected to have some basic knowledge of algebra, topology and category theory, however, all recurrent categorical concepts are included in a preliminary chapter.
