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Relational Topology

Authors: Univ.-Prof. Dr. Gunther Schmidt, Prof. Dr. Michael Winter

Publisher: Springer International Publishing

Book Series : Lecture Notes in Mathematics

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

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About this book

This book introduces and develops new algebraic methods to work with relations, often conceived as Boolean matrices, and applies them to topology. Although these objects mirror the matrices that appear throughout mathematics, numerics, statistics, engineering, and elsewhere, the methods used to work with them are much less well known. In addition to their purely topological applications, the volume also details how the techniques may be successfully applied to spatial reasoning and to logics of computer science.

Topologists will find several familiar concepts presented in a concise and algebraically manipulable form which is far more condensed than usual, but visualized via represented relations and thus readily graspable. This approach also offers the possibility of handling topological problems using proof assistants.

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Table of Contents

Frontmatter

Chapter 1. Introduction

Abstract

There exist lots of concepts around topology: open sets, neighborhoods, transitions to their open kernels, proximity, nearness, betweenness, apartness, different concepts of contact and so on. Although, they are all heavily interrelated, this is often hard to recognize, because they are discussed in quite different settings resp. terminology. We are going to identify the core concepts of those ideas and to show how they may be mutually deduced from one another.

Gunther Schmidt, Michael Winter

Chapter 2. Prerequisites

Abstract

Relational methods are not yet broadly known and, thus, need a detailed introduction. We develop all the necessary methodology; it originates in particular from Schmidt and Ströhlein (Relationen und Graphen. Mathematik für Informatiker. Springer, 1989; Relations and graphs—discrete mathematics for computer scientists. EATCS monographs on theoretical computer science. Springer, 1993), Schmidt (Relational mathematics. Encyclopedia of mathematics and its applications, vol 132. Cambridge University Press, 2011, 584 pp), Schmidt and Winter (Relational Mathematics continued. Technical Report 2014-01, Fakultät für Informatik, Universität der Bundeswehr München, April 2014). There, full proofs may be found. In addition it is shown how everything is based on a concise axiomatic basis. However, some of the following results are new, and therefore given together with their proof.

Gunther Schmidt, Michael Winter

Chapter 3. Products of Relations

Abstract

In Definition 2.2.1, we have introduced the direct power of a set—modelling the concept of a powerset—and shown that it is uniquely determined up to isomorphism. Even earlier, we have defined the natural projection of a set equipped with an equivalence to the set of its classes. We are now going to handle the direct product and direct sum.

Gunther Schmidt, Michael Winter

Chapter 4. Meet and Join as Relations

Abstract

When, in the preceding chapter, we had pairs (and so iterated also tuples), we immediately proceeded to handling binary mappings with relational means. This concerned the very general concepts such as being commutative, distributive, or associative. A more specific law concerns absorption mainly occurring in one traditional environment, namely for binary meets and joins. They will be handled here accordingly when the following cone mappings are available.

Gunther Schmidt, Michael Winter

Chapter 5. Applying Relations in Topology

Abstract

Since its first appearence in the book Vorstudien zur Topologie by Johann Benedict Listing of 1847, topology (then and for a long period termed analysis situs ) has been given many facets; among the main ones are considerations of neighborhoods, open sets, and closed sets. We start here, giving the corresponding definitions lifted to point-free as well as quantifier-free versions, showing how they are interrelated, thus exhibiting their cryptomorphism and offering the possibility to transform one version into the other, not least visualizing them via TituRel programs.

Gunther Schmidt, Michael Winter

Chapter 6. Construction of Topologies

Abstract

We investigate three frequently applied methods of constructing a topology from other given topologies, namely the relative topology, the quotient topology, and the product topology.

Gunther Schmidt, Michael Winter

Chapter 7. Closures and Their Aumann Contacts

Abstract

Topology has been shown to be definable in several cryptomorphically equivalent ways: by a neighborhood system, by a collection of open sets (be these given as a vector along the powerset or as a partial diagonal on it), by a collection of closed sets, or by a mapping to open kernels.

Gunther Schmidt, Michael Winter

Chapter 8. Proximity and Nearness

Abstract

Proximity is introduced when trying to axiomatize the concept of being in some sense “near” that may hold from a set to another set. Far better known are point-to-set notions that characterize being element of a neighborhood or of an open set. The first concept of proximity was described in 1908 by Frigyes Riesz and then ignored. Others to be mentioned for having worked on such ideas include V. A. Efremovič in 1934 and A. N. Wallace in 1940. More recently, we found some work in Naimpally and Warrack (Proximity Spaces, Cambridge University Press, 1970), Vakarelov et al. (J Appl Non-Class Log 12:527–559, 2002), Bennett and Düntsch (Axioms, Algebras and Topology. In Marco Aiello, Ian E. Pratt-Hartmann, and Johan F.A.K. van Bentham, editors, Handbook of Spatial Logics, pages 99–159.Springer, 2007).

Gunther Schmidt, Michael Winter

Chapter 9. Frames

Abstract

There exists a scenario in computer science where intricate topological questions are discussed. The topic is best described considering a device we observe without any knowledge about its inner program or process structure. This means necessarily incomplete observations which are somehow ordered by precision. Handling such observations requires specific orderings and often entails employing topological concepts.

Gunther Schmidt, Michael Winter

Chapter 10. Simplicial Complexes

Abstract

This section is intended to show how one might work relationally also for algebraic topology. We give a glimpse of simplicial complexes, usually subsumed under that topic.

Gunther Schmidt, Michael Winter

Backmatter

Title: Relational Topology
Authors: Univ.-Prof. Dr. Gunther Schmidt
Prof. Dr. Michael Winter
Copyright Year: 2018
Publisher: Springer International Publishing
Electronic ISBN: 978-3-319-74451-3
Print ISBN: 978-3-319-74450-6
DOI: https://doi.org/10.1007/978-3-319-74451-3

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