Topology with Applications: Topological Spaces via Near and Far

294 Pages·2013·2.54 MB·English

TOPOLOGY WITH
APPLICATIONS
Topological Spaces via Near and Far
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TOPOLOGY WITH
APPLICATIONS
Topological Spaces
via Near and Far
Somashekhar A. Naimpally
Lakehead University, Canada
James F. Peters
University of Manitoba, Canada
World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI
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TOPOLOGY WITH APPLICATIONS
Topological Spaces via Near and Far
Copyright © 2013 by World Scientific Publishing Co. Pte. Ltd.
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ISBN 978-981-4407-65-6
Printed in Singapore.
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January15,2013 9:54 WorldScientificBook-9inx6in TopologyApplications
We dedicate this book to
Sudha and Sheela and Saroja
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Foreword
Topological ideas originated in the 19th century, mainly in the works of
Riemann and Poincaré, and finally led to the classical definitions of topo-
logical space either via the closure operator by Kuratowski, or via open
sets. The latter one is the most popular now. However, such progress was
precededbymanyotherattemptsandapproacheswhichappeartobeuseful
in different cases and worthwhile to be developed.
One such approach is based on the notion of near sets introduced by
F. Riesz in 1908. A set X with a nearness relation between its subsets is
calleda proximityspace and everysuchstructure induces a topologyon X
definedviathe closureoperator: wesaythata pointxlies inthe closureof
{ }
a subset U if the subset x is near U. It appears that the same topology
on X may correspond in this way to different proximities.
Moreover many topological results may be inherited from statements
concerningproximityspaces. Ithastobe recalledthatinthe samemanner
the proximity structure is induced by the uniform relation introduced by
A. Weil.
The proximity was rediscovered by V.A. Efremovič in the middle of
1930s and later A.D. Wallace arrived at the same concept. Initially, the
proximity theory has developed by its own means with strong contribu-
tions by Efremovič and his school and the first author of this book, S.A.
Naimpally.
Recently, the proximity theory found valuable applications which
strongly justify their developments. This book was preceded by a short
1
excellentsurvey by the same authorsofthe mainideasofnearsets,which
1
Peters, Jim; Naimpally, Som. Applications of near sets. Notices Amer. Math. Soc.
59(2012), no. 4,536–542.
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viii Topology with Applications. Topological Spaces via Near and Far
haveimportantapplicationsinimageprocessing,imageanalysis,facerecog-
nition, and some other engineering and natural science problems.
In a sense, the way in which this theory came from its mathematical
origin to applications is quite opposite to the development of fuzzy sets
arising from applications and leading to a branch of modern set theory.
The targets of this book are three-fold. First, it is to expose the main
constructions and ideas of set-theoreticaltopology throughthe magnifying
glass of proximity and uniform spaces. Such an introduction to topology
strongly benefits from the simple as well as rigorous notions of near and
far at various levels. The motivation originates in advanced calculus and
goes to metric spaces, topology, proximity, and uniformity. Diverse results
are unified, as for example, in the proximal avatar of the result by A.D.
Taimanov on extensions of continuous functions from dense subsets.
Second, this book gives an introduction to descriptive proximity rising
from extensions of the usual spatial nearness relations in terms of descrip-
tivelynearsets. Thisleadstomanyimportantresultsthatarefundamental
building blocks in understanding topology and its implications.
Third, this book demonstratesthe utility of the notions of near and far
in many diverse applications, ranging from cell biology and micropalaeon-
tology to camouflage and forgery detection. The applications are made
incisive by the accompanying use of near and far in bringing to light the
subtleties typically hidden in physical structures such as the distinctively
differentshapesofmicrofossilsandtypicalcamouflageofanimalsinnatural
surroundings. Ofparticularinterestinthisbookistherangeofapplications
of the proposed approach to topology in study of climate change, mineral
and fossil fuel exploration.
We strongly recommend this book as a concise and very original in-
troduction of set-theoretical topology, nearness theory and their modern
applications.
October 2012 Professor Iskander A. Taimanov, D. Sc.,
Member of Russian Academy of Sciences,
Chair of Geometry and Topology,
Novosibirsk State University, Russia
January15,2013 9:54 WorldScientificBook-9inx6in TopologyApplications
Preface
The main purpose inwriting this book is to demonstratethe beneficialuse
of near and far, discovered by F. Riesz over a century ago, from the un-
dergraduate to the research level in general topology and its applications.
Use of near and far is intuitive yet rigourous at the same time, which is
rareinmathematics. Thenear andfar paradigmisbasedonmanyyearsof
teaching and research by the authors. The book introduces topology and
its many applications viewed within a framework that begins with met-
ric spaces and deals with the usual topics such as continuity, open and
closedsets,metric nearness,compactnessand completenessand glidesinto
topology, proximity and uniformity. Most topics are first studied in metric
spaces and later in a topological space. The motivation for this approach
is a straightforward, intuitive and yet rigourous rendition of topological
concepts. The approach also unifies several scattered results in many ar-
eas. Many exercises come from the current literature and some occur in
simplified form in metric spaces. The end result is a solid, workableframe-
work for the study of topology with a variety of applications in science
and engineering that include camouflage, cell biology, digital image anal-
ysis, forgery detection, general relativity, microscopy, micropalaeontology,
pattern recognition,population dynamics, psychologyandvisualmerchan-
dising.
We gratefully acknowledge the insightful Foreword by I.A. Taimanov.
This Foreword is especially important, since it lucidly brings together the
principal highlights of this book and it serves as a commemoration of the
seminal work on topology by A.D. Taimanov, who proved one of the most
fundamental results in topology concerning extensions of continuous func-
tions from dense subspaces.
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