The featured review of the AMS describes the author's earlier work in
the field of approach spaces as, 'A landmark in the history of general
topology'. In this book, the author has expanded this study further and
taken it in a new and exciting direction.
The number of conceptually and technically different systems which
characterize approach spaces is increased and moreover their uniform
counterpart, uniform gauge spaces, is put into the picture. An extensive
study of completions, both for approach spaces and for uniform gauge
spaces, as well as compactifications for approach spaces is performed. A
paradigm shift is created by the new concept of index analysis.
Making use of the rich intrinsic quantitative information present in
approach structures, a technique is developed whereby indices are
defined that measure the extent to which properties hold, and theorems
become inequalities involving indices; therefore vastly extending the
realm of applicability of many classical results. The theory is then
illustrated in such varied fields as topology, functional analysis,
probability theory, hyperspace theory and domain theory. Finally a
comprehensive analysis is made concerning the categorical aspects of the
theory and its links with other topological categories.
Index Analysis will be useful for mathematicians working in category
theory, topology, probability and statistics, functional analysis, and
theoretical computer science.
