Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top

2015 | Book

Approach Spaces Read chapter Read first chapter

Index Analysis

Approach Theory at Work

Author: R. Lowen

Publisher: Springer London

Book Series : Springer Monographs in Mathematics

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

Login to get access
insite
SEARCH

About this book

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.

Table of Contents

Frontmatter
Chapter 1. Approach Spaces
Abstract
In this first chapter we define the basic structures which determine what is called an approach space. One of the powerful features of approach spaces is that they can be determined by no less than 10 conceptually totally different but nevertheless equivalent structures. These can have a topological side and/or a metric side to them and the reason for is made clear in the second chapter.
R. Lowen
Chapter 2. Topological and Metric Approach Spaces
Abstract
Both topological and metric spaces can be viewed as special types of approach spaces. More precisely, both the categories of topological spaces and continuous maps, and of (quasi)-metric spaces and nonexpansive maps, can be embedded as full and isomorphism-closed subcategories of the category of approach spaces, the former as a stable (i.e. simultaneously concretely reflective and concretely coreflective) subcategory and the latter as a concretely coreflective subcategory.
R. Lowen
Chapter 3. Approach Invariants
Abstract
In view of the fact that approach spaces generalize at the same time topological spaces and metric spaces, there will be approach properties or invariants which are more of a topological nature while others are more of a metric nature. We do not systematically generalize all possible concepts in topological and metric spaces to the setting of approach spaces but concentrate ourselves on those concepts which demonstrate some interesting aspects, appear to live naturally in the realm of approach theory or which are required, in particular, for the applications later on.
R. Lowen
Chapter 4. Index Analysis
Abstract
Rather than being restricted to asserting that a space, or a subset of a space, or a function, or any other item defined in terms of the structures at hand, does or does not have a certain property approach theory provides us with a canonical machinery by means of which we can define numerical indices of properties. The smaller an index is the better the property is approximated. In this chapter it becomes abundantly clear that the systematic use of indices lies at the heart of approach theory as they are built into the basics of the theory.
R. Lowen
Chapter 5. Uniform Gauge Spaces
Abstract
Approach spaces form a local theory. Nevertheless there is also a natural uniform notion of completeness and completion and in this chapter we define the appropriate setting hereto, namely the category of uniform gauge spaces. This setting is linked to uniform spaces. Just as was the case for approach spaces here this category will be a supercategory of both the categories of uniform spaces and uniformly continuous maps and of metric spaces with non-expansive maps. Again, the former is a stable subcategory and the latter is a concretely coreflective subcategory. Finally we also consider the non-symmetric variant, quasi-uniform gauge spaces.
R. Lowen
Chapter 6. Extensions of Spaces and Morphisms
Abstract
The fact that approach spaces generalize at the same time topological and metric spaces implies that we can consider at the same time both a notion of completion and a notion of compactification, especially for uniform approach spaces. Further we also consider a completion for uniform gauge spaces. All these extensions turn out to be epireflections in their respective settings and hence also give rise to natural extensions of morphisms.
R. Lowen
Chapter 7. Approach Theory Meets Topology
Abstract
In Chaps. 7–11 we study applications of approach theory to various areas of mathematics. Here we highlight two areas in topology, namely function spaces and the Cech-Stone compactification. In the first section we treat function spaces, starting from arbitrary approach spaces and uniform gauge spaces. The results are applicable to any situation where for instance a choice has been made of a particular metric for a topology or a uniformity. In the second section we study the approach version of the Cech-Stone compactification of an Atsuji space, and in particular of the natural numbers, in more detail.
R. Lowen
Chapter 8. Approach Theory Meets Functional Analysis
Abstract
Here we start in the classical setting of normed spaces and see how the well-known weak and weak* topologies can be quantified by canonical approach structures and how using index analysis, this allows to obtain quantified results of which several classical results are simple corollaries. In a second part we see that the construction of the weak and weak* approach structures on normed spaces fits into the wider picture of what we call approach vector spaces and locally convex approach spaces.
R. Lowen
Chapter 9. Approach Theory Meets Probability
Abstract
Here we see that the construction of the weak* approach structure performed in the previous chapter, as in the classical topological setup, when restricted to probability measures allows for a quantification of the weak topology on probability measures. However we also consider other quantifications, depending on the problem at hand. In the first section we consider the general case of probability measures on a Polish space, in the second section we consider a quantification of convergence in probability for random variables. In the last section we prove an indexed version of the Lindeberg-Feller central limit theorem making use of a natural Lindeberg index indicating to what extent the Lindeberg condition is fulfilled.
R. Lowen
Chapter 10. Approach Theory Meets Hyperspaces
Abstract
Here we study approach structures (or uniform gauge structures) on hyperspaces of closed sets of metric spaces. In the first section we study a natural quantification of the Wijsman topology, in the second section we study the proximal topologies and in the last example, we study a quantified version of the Vietoris structure in the more general setup of closed sets in an arbitrary approach space.
R. Lowen
Chapter 11. Approach Theory Meets DCPO’s and Domains
Abstract
Here we propose an intrinsic solution for the problem of quantifiability in domain theory. We construct an approach structure which quantifies the Scott topology regardless of cardinality conditions on bases. We show that every domain is quantifiable in this sense. We get weightability for free and in the case of an algebraic domain satisfying the Lawson condition, a quantifying approach space can be obtained with a weight satisfying the so-called kernel condition. With respect to contractions, we study fixed point theorems. For monotone as well as for non-monotone maps we establish new fixed point theorems and we recover several existing ones as special cases.
R. Lowen
Chapter 12. Categorical Considerations
Abstract
In this final chapter we go back to basics and treat several different categorical aspects of the theory of approach spaces. First, we establish the existence of an infinite collection of stable subcategories of the category approach spaces. Second, we construct a quasi-topos supercategory of the category of approach spaces which will serve as a starting-point for the construction of all the hulls. We then construct the extensional topological hull, the quasi-topos hull and the cartesian closed topological hull. Finally we give a new proof of a lax-algebraic description of the category of approach spaces.
R. Lowen
Backmatter
Metadata
Title
Index Analysis
Author
R. Lowen
Copyright Year
2015
Publisher
Springer London
Electronic ISBN
978-1-4471-6485-2
Print ISBN
978-1-4471-6484-5
DOI
https://doi.org/10.1007/978-1-4471-6485-2

Premium Partner

    Image Credits