Lowen R. Index Analysis: Approach Theory at Work

Springer, 2015. — 466 p. — (Springer Monographs in Mathematics) — ISBN: 1447164849, 9781447164845The 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.Structures and Theories.
Languages and Structures.
Theories.
Definable Sets and Interpretability
Basic Techniques.
The Compactness Theorem.
Complete Theories.
Up and Down.
Back and Forth.
Algebraic Examples.
Quantifier Elimination.
Algebraically Closed Fields.
Real Closed Fields.
Realizing and Omitting Types.
Omitting Types and Prime Models.
Saturated and Homogeneous Models.
The Number of Countable Models.
Indiscernibles.
Partition Theorems.
Order Indiscernibles.
A Many-Models Theorem.
An Independence Result in Arithmetic.
ω-Stable Theories.
Uncountably Categorical Theories.
Morley Rank.
Forking and Independence.
Uniqueness of Prime Model Extensions.
Morley Sequences.
ω-Stable Groups.
The Descending Chain Condition.
Generic Types.
The Indecomposability Theorem.
Definable Groups in Algebraically Closed Fields.
Finding a Group.
Geometry of Strongly Minimal Sets.
Pregeometries.
Canonical Bases and Families of Plane Curves.
Geometry and Algebra.
Set Theory.
Real Algebra.