# Index Analysis

## Approach Theory at Work

Part of the Springer Monographs in Mathematics book series (SMM)

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Part of the Springer Monographs in Mathematics book series (SMM)

A 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.

Approach Space Ascoli Theorem Asymptotic Center Asymptotic Radius Completion Contraction Dini’s Theorem Distance Index Limit Operator Measure of Non-Compactness Metric Metric Space Opial’s Condition Proximity Topological Space Topology Uniform Gauge Space Uniform Space Uniformity

- DOI https://doi.org/10.1007/978-1-4471-6485-2
- Copyright Information Springer-Verlag London 2015
- Publisher Name Springer, London
- eBook Packages Mathematics and Statistics
- Print ISBN 978-1-4471-6484-5
- Online ISBN 978-1-4471-6485-2
- Series Print ISSN 1439-7382
- Series Online ISSN 2196-9922
- Buy this book on publisher's site