Non-Hausdorff Topology and Domain Theory: Selected Topics in Point-Set Topology

500 Pages·2013·2.894 MB·English

by  Jean Goubault-Larrecq

more information – www.cambridge.org/9781107034136

Non-HausdorffTopologyandDomainTheory
This unique book on modern topology looks well beyond traditional treatises, and
explores spaces that may, but need not, be Hausdorff. This is essential for domain
theory, the cornerstone of semantics of computer languages, where the Scott topol-
ogyisalmostneverHausdorff.Forthefirsttimeinasinglevolume,thisbookcovers
basic material on metric and topological spaces, advanced material on complete par-
tialorders,Stoneduality,stablecompactness,quasi-metricspaces,andmuchmore.An
early chapter on metric spaces serves as an invitation to the topic (continuity, limits,
compactness,completeness)andformsacompleteintroductorycoursebyitself.
Graduate students and researchers alike will enjoy exploring this treasure trove of
results.Fullproofsaregiven,aswellasmotivatingideas,clearexplanations,illuminat-
ingexamples,applicationexercises,andsomemorechallengingproblemsforadvanced
readers.
JEAN GOUBAULT-LARRECQ is Full Professor of Computer Science at Ecole
Normale Supérieure de Cachan, France. He obtained his Ph.D. in 1993 from Ecole
Polytechniqueinthefieldofautomateddeduction,andsincethenhehashadanactive
careerinseveralfieldsofcomputerscience:logic,computersecurity,semantics,dom-
ain theory, and probabilistic and non-deterministic systems. He is currently heading
teamSECSI(securityofinformationsystems)atINRIA,France’snationalinstitutefor
researchincomputerscienceandcontrol.Heistherecipientofthe2011CNRSSilver
Medalinthefieldofcomputerscienceanditsinteractions.Thisisthehighestscientific
distinctionincomputerscienceinFrance.
NEW MATHEMATICAL MONOGRAPHS
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Non-Hausdorff Topology
and Domain Theory
JEAN GOUBAULT-LARRECQ
EcoleNormaleSupérieuredeCachan,France
CAMBRIDGE UNIVERSITY PRESS
Cambridge,NewYork,Melbourne,Madrid,CapeTown,
Singapore,SãoPaulo,Delhi,MexicoCity
CambridgeUniversityPress
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www.cambridge.org
Informationonthistitle:www.cambridge.org/9781107034136
©JeanGoubault-Larrecq2013
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Firstpublished2013
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Contents
1 Introduction 1
2 Elementsofsettheory 4
2.1 Foundations 4
2.2 Finiteness,countability 6
2.3 Ordertheory 8
2.4 TheAxiomofChoice 15
3 Afirsttouroftopology:metricspaces 18
3.1 Metricspaces 18
3.2 Convergence,limits 20
3.3 Compactsubsets 25
3.4 Completemetricspaces 33
3.5 Continuousfunctions 37
4 Topology 46
4.1 Topology,topologicalspaces 46
4.2 Orderandtopology 53
4.3 Continuity 62
4.4 Compactness 67
4.5 Products 74
4.6 Coproducts 80
4.7 Convergenceandlimits 82
4.8 Localcompactness 91
4.9 Subspaces 95
4.10 Homeomorphisms,embeddings,quotients,retracts 97
4.11 Connectedness 106
4.12 AbitofcategorytheoryI 111
5 Approximationandfunctionspaces 120
5.1 Theway-belowrelation 120
5.2 Thelatticeofopensubsetsofaspace 140
v
vi Contents
5.3 Spacesofcontinuousmaps 149
5.4 Theexponentialtopology 151
5.5 AbitofcategorytheoryII 169
5.6 C-generatedspaces 180
5.7 bc-domains 194
6 Metrics,quasi-metrics,hemi-metrics 203
6.1 Metrics,hemi-metrics,andopenballs 203
6.2 ContinuousandLipschitzmaps 209
6.3 Topologicalequivalence,hemi-metrizability,metrizability 214
6.4 Coproducts,quotients 236
6.5 Products,subspaces 239
6.6 Functionspaces 241
6.7 Compactnessandsymcompactness 251
7 Completeness 260
7.1 Limits,d-limits,andCauchynets 260
7.2 Astrongformofcompleteness:Smyth-completeness 267
7.3 Formalballs 279
7.4 Aweakformofcompleteness:Yoneda-completeness 288
7.5 Theformalballcompletion 311
7.6 Choquet-completeness 323
7.7 Polishspaces 334
8 Soberspaces 341
8.1 FramesandStoneduality 341
8.2 Soberspacesandsobrification 351
8.3 TheHofmann–MisloveTheorem 364
8.4 Colimitsandlimitsofsoberspaces 383
9 Stablycompactspacesandcompactpospaces 397
9.1 Stablylocallycompactspaces,stablycompactspaces 397
9.2 Coproductsandretractsofstablycompactspaces 411
9.3 Productsandsubspacesofstablycompactspaces 413
9.4 Patch-continuous,perfectmaps 419
9.5 Spectralspaces 423
9.6 Bifinitedomains 440
9.7 Noetherianspaces 453
References 475
Notationindex 480
Index 484
1
Introduction
Purpose This book is an introduction to some of the basic concepts of
topology,especiallyofnon-Hausdorff topology.Iwillofcourseexplainwhat
it means (Definition 4.1.12). The important point is that traditional topology
textbooksassumetheHausdorffseparationconditionfromtheverystart,and
contain very little information on non-Hausdorff spaces. But the latter are
importantalreadyinalgebraicgeometry,andcrucialinfieldssuchasdomain
theory.
Conversely,domaintheory(AbramskyandJung,1994;Gierzetal.,2003),
whicharosefromlogicandcomputerscience,startedasanoutgrowthofthe-
ories of order. Progress in this domain rapidly required a lot of material on
(non-Hausdorff)topologies.
After about 40 years of domain theory, one is forced to recognize that
topology and domain theory have been beneficial to each other. I’ve already
mentioned what domain theory owes to topology. Conversely, in several
respects,domaintheory,inabroadsense,istopologydoneright.
This book is an introduction to both fields, dealt with as one. This seems
to fill a gap in the literature, while bringing them forth in a refreshing
perspective.
Secondarypurpose Thisbookisself-contained.Mymaininterest,though,as
anauthor,wastoproduceauniquereferenceforthekindofresultsintopology
and domain theory that I needed in research I started in 2004, on semantic
modelsofmixednon-deterministicandprobabilisticchoice.Thegoalquickly
grewoutofproportion,andwillthereforeoccupyseveralvolumes.Thecurrent
book can be seen as the preliminaries for other books on these other topics.
Some colleagues of mine, starting with Professor Alain Finkel, have stressed
that these preliminaries are worthy of interest, independently of any specific
application.
1
2 Introduction
Teaching This book is meant to be used as a reference. My hope is that it
shouldbeusefultoresearchersandtopeoplewhoarecurioustoreadabouta
modernviewofpoint-settopology.Itdidnotariseaslecturenotes,andIdon’t
thinkitcanbeusedassuchdirectly.Ifyouplanonusingthisbookasabasis
for lectures, you should extract a few selected topics. Think of yourself as a
scriptwriterandthisbookasanovel,andpretendthatyourjobistoproducea
shorterscriptforafeature-lengthmovie,andskipthelessimportantsubplots.
Exercises Therearemanyexercises,spreadoverthewholetext.Someofthem
aremeantfortraining,i.e.,tohelpyouunderstandnotionsbetter,tomakeyou
morecomfortablewithdefinitionsandtheoremsthatwehavejustseen.Some
othersaretheretohelpyouunderstandnotionsinadeeperway,ortogofurther.
Thisistraditionalinmathematicaltextbooks.Itisthereforeprofitabletoread
allexercises:thoseofthesecondkindinparticularstateadditionaltheorems,
whichyoucanproveforyourself(sometimeswiththehelpofhints),butdonot
needto.Itwillhappenthatnotonlysolutionsofexercisesbutalsosomeproofs
of theorems will depend on results we shall have seen in previous exercises.
Thereisnopressureonyoutoactuallydoanyexercise,andyoucandecideto
takethemasameresourceofadditionalinformation.
What is not covered Topology is an extremely rich topic, and I could not
coverallsubtopicsinabookofreasonablesize.(Notwithstandingthefactthat
Icertainlydonotknoweverythingintopology.)Idecidedtomakeaselection
among those subtopics that pleased me most. Some other topics were neces-
sarilyleftout,despitethembeingequallyinteresting.Forexample,algebraic
topology willnot be touched upon at all.Uniform and quasi-uniform spaces,
bitopological spaces, Lindelöf spaces, Souslin spaces, and topological group
theorywereleftoutaswell.Topologicalconvexity,topologicalmeasuretheory,
hyperspaces,andpowerdomainswillbetreatedinfurthervolumes.
Dependencies Readingu ←vas“vdependsonu,”thestructureofthebook
isasfollows.Dashedarrowsmarkweakerdependencies.