Mappings between distance sets or spaces [Elektronische Ressource] / von Jobst Heitzig

Dipl.-Math. Jobst Heitzig

Sujets

• Mathematics

Informations

• Publié par: gottfried_wilhelm_leibniz_universitat_hannover
• Publié le: 01 janvier 2003
• Nombre de lectures: 47
• Langue: Deutsch
• Poids de l'ouvrage: 1 Mo

Extrait

Mappings
betweendistancesetsorspaces
VomFachbereichMathematik
derUniversit tHannover
zurErlangungdesGrades
DoktorderNaturwissenschaften
Dr.rer.nat.
genehmigteDissertation
von
Dipl.-Math.JobstHeitzig
geborenam17.Juli1972inHannover
2003
diss.tex; 17/02/2003; 8:34; p.1Referent:Prof.Dr.MarcelErnØ
Korreferent:Prof.Dr.Hans-PeterK nzi
TagderPromotion:16.Mai2002
diss.tex; 17/02/2003; 8:34; p.2Zusammenfassung
Die vorliegende Arbeit hat drei Ziele: Distanzfunktionen als wichtiges Werkzeug der
allgemeinen Topologie wiedereinzuf hren; den Gebrauch von Distanzfunktionen auf
den verschiedensten mathematischen Objekten und eine Denkweise in Begriffen der
Abstandstheorie anzuregen; und schliesslich spezifischere Beitr ge zu leisten durch die
CharakterisierungwichtigerKlassenvonAbbildungenunddieVerallgemeinerungeiniger
topologischerS tze.
Zun chst werden die Konzepte des ,Formelerhalts‘ und der , bersetzung von
Abst nden‘ benutzt, um interessante ,nicht-topologische‘ Klassen von Abbildungen zu
finden, was zur Charakterisierung vieler bekannter Arten von Abbildungen mithilfe von
Abstandsfunktionenf hrt.Nachdemdanneine,kanonische‘MethodezurKonstruktion
von Distanzfunktionen angegeben wird, entwickele ich einen geeigneten Begriff von
,Distanzr umen‘, der allgemein genug ist, um die meisten topologischen Strukturen
induzieren zu k nnen. Sodann werden gewisse Zusammenh nge zwischen einigen
Arten von Abbildungen bewiesen, wie z.B. dem neuen Konzept ,streng gleichm ssiger
Stetigkeit‘. Es folgt eine neuartige Charakterisierung der ˜hnlichkeitsabbildungen zwis-
chen Euklidischen R umen. Die Dissertation schliesst mit einigen Verallgemeinerungen
bekannter Vervollst ndigungskonstruktionen und wichtiger Fixpunkts tze, und einer
kurzenStudie berTechnikenderVisualisierungvonAbst nden.
Abstract
The aim of this thesis is threefold: to reinstate distance functions as a principal tool
of general topology; to promote the use of distance functions on various mathematical
objects and a thinking in terms of distances also in non-topological contexts; and to
make more specific contributions by characterizing important classes of mappings and
generalizingsomeimportanttopologicalresults.
I start by using the key concepts of ‘preservation of formulae’ and ‘translation of
distances’toextractinteresting‘non-topological’classesofmappings,whichleadstothe
characterization of many well-known types of mappings in terms of distance functions.
Aftergivinga‘canonical’methodforconstructingdistancefunctions,asuitablenotionof
‘distancespaces’willbedeveloped,generalenoughtoinducemosttopologicalstructures.
Then certain relationships between many kinds of mappings are proved, including the
new concept of ‘strong uniform continuity’, followed by a new characterization of the
similarity maps between Euclidean spaces. The thesis closes with some generalizations
of completions and fixed point theorems, and a short, self-contained study of distance
visualizationtechniques.
Schlagworte:Distanzfunktion,Abbildung,Topologie
Keywords:distancefunction,mapping,topology
diss.tex; 17/02/2003; 8:34; p.3Danksagung
Dank geht an Herrn Dipl.-Math. Lars Ritter und Dr. Daniel Frohn f r
Diskussionen und gelegentliche Korrekturlesearbeiten, und an Dr. J rgen
Reinholdf rdieerfolgreichegemeinsameZ hlerei.Dr.MatthiasKrieselldanke
ich sehr f r die Bereitschaft, sich immer wieder auch unausgegorene Ideen
anzuh renundmeinenGehirnknotenweitereKantenhinzuzuf gen.
Vor allen anderen gilt mein Dank meinem Doktorvater Prof.Dr. Marcel
TonioErnØ,mitdemichunz hlbareStundenfruchtbarenGedankenaustauschs
teilendurfteunddemesgelang,unerm dlichmeinenmathematischenHorizont
zu erweitern. Ohne seine Unterst tzung w re eine solche breitangelegte Studie
wohlkaumm glichgewesen.
diss.tex; 17/02/2003; 8:34; p.4Contents
AGENERALCONCEPTOFDISTANCE
1.Distancesets 6
Definitions 6 Real distances 7 Multi-real distances 12
Distances in classical algebraic structures 15 Some other concepts of ‘generalized metric’ 23
2.Mappings 25
Distance sets with the same value monoid 25 Translating distances of different type 29
Comparing distance functions on the same set 40
BACKTOTHEROOTSOFTOPOLOGY
3.Convergenceandclosure 48
Converging to the right class of structures 48 Open and closed; filters and nets 54
Distances in point-free situations, and hyperspaces 62
4.Moreonmappings 68
Topological properties of maps 68 Maps with both topological and non-topological properties 72
Finest distance structures 82
5.Fundamentalnetsandcompleteness 96 nets and Cauchy-filters 96 Notions of completeness 99 Distances between nets 102
Some completions 105
6.Fixedpoints 110
Banach: Fixed points of Lipschitz-continuous maps 110
Sine Soardi: Fixed points of contractive maps 114 Brouwer: Fixed points of continuous maps 115
APPENDIX
Visualizationofdistances 122
Additionalproofs 137
References 140
Indices 144
diss.tex; 17/02/2003; 8:34; p.5diss.tex; 17/02/2003; 8:34; p.6Railroad, telephone, bicycle, automobile, air plane,
and cinema revolutionized the sense of distance. [:::]
Distances depended on the effect of memory, the force of
emotions, and the passage of time.
Stephen Kern,
The Culture Of Time And Space 1880 1918
In everyday language, ‘distance’ has always been something more general than
thelengthofasegmentinsomegeometricalspace.Instead,theconceptof‘near’
and ‘far’ is one of the more important categories in human thinking. Extracting
the abstract idea from the physical phenomenon, we speak of the growing
distance to an old friend, of how near we are to reach a certain goal, or how
farfrombeingjealous.Itisimportantthatquiteoftentheinterestingquestionis
not ‘‘how much is in betweenx andy’’ but rather ‘‘how much is needed to get
from x to y’’. This somewhat dynamical interpretation of distance differs from
thegeometrical oneinthatitdoesnotimplyanysymmetry,positivity,orstrictness
a priori.
It is only natural when mathematicians, too, think of their objects as being
relatedbytheoneorotherkindofdistance andhowsurprisingisitthatwestill
require mathematical distances to be real-valued, mostly symmetric, and non-
negative?Before1900,mathematicaldistanceshadbeedusedmainlyingeometry
and as a measure of difference between real numbers or functions. They had
alsoplayedanimportantrolefortheclarificationofthenotionof‘realnumber’
itself, which in turn was a strong impetus for the development of topology.
In the beginning of the last century, when FrØchet [FrØ05, FrØ06, FrØ28] and
Hausdorff [Hau14, Hau27, Hau49] initiated the axiomatic study of distances in
thegeneralsettingofmetricinsteadofgeometricspaces,therealnumberswere
1
ODUCTIONINTR2
therefore the natural candidates for the values of a distance function. Complex
numbers or real vectors, being imaginable alternatives, would most certainly
not have been considered suitable because of the difficulties in ordering such
entities giventhatpartialordershadnotreceivedmuchattentionatthattime.
On the other hand, rational numbers had already long been known to be too
specialbecauseoftheirlackingcompleteness.Justasincaseofmeasuretheory,
it is therefore not surprising that the theoretical treatment of distances was
dominatedbyaparadigmofusingrealnumbers.
Although,fromthebeginning,generaltopologywasfarmorethanthestudy
of metric spaces, the question of which topological spaces can be endowed
withasuitablemetric,knownasthe‘metrizationproblem’,remainedimportant.
This was not only because metric spaces had very nice topological properties,
mostly inherited from even nicer properties of the real numbers themselves,
but also since the idea of distance remained a principal intuition in building
newtopologicalconcepts,andbecausetopologicalspacesalonehadnotenough
structure to formulate certain interesting notions. For example, Lipschitz- and
uniform continuity, or completeness, being of great importance in real analysis,
cannotbeexpressedintermsofopensetsalone.
This motivated the search for suitable structural additives to general topo-
logical spaces, which could well have led to an early study of substantially
more general distance functions than real metrics. But despite only a few
attempts in the latter direction, the researchers in this field soon focused on
systems of subsets instead, ending up with the notion of ‘uniform space’ (cf.
[BHH98]). However, there were situations when distances had a great chance
of being reconsidered passing virtually unnoticed. Van Dantzig [vD32], for
instance,definedfundamentalsequencesinatopologicalgroup,usingMenger’s
‘Gruppenmetrik’ [Men31] without recognizing it as a distance function. Even
more surprisingly, Kelley essentially proved that every uniformity (even every
quasi-uniformity)comesfromafamilyofreal-valueddistancefunctions[Kel55],
butdespitethepopularityofhisclassicaltextbook,thetheoryofuniformspaces
did not yet enter a possibly fruitful engagement with a theory of vector-valued
metrics.
The aim of this thesis is threefold: to reinstate distance functions as a principal
tool of general topology; to promote the use of distance on various
mathematicalobjectsandathinkingintermsofdistancesalsoinnon-topological
contexts; and to make more specific contributions by characterizing important
classesofmappingsandgenerali