A Cp-Theory Problem Book: Compactness in Function Spaces PDF
Preview A Cp-Theory Problem Book: Compactness in Function Spaces
Problem Books in Mathematics Vladimir V. Tkachuk A Cp-Theory Problem Book Compactness in Function Spaces Problem Books in Mathematics SeriesEditor: PeterWinkler DepartmentofMathematics DartmouthCollege Hanover,NH03755 USA Moreinformationaboutthisseriesathttp://www.springer.com/series/714 Vladimir V. Tkachuk A Cp-Theory Problem Book Compactness in Function Spaces 123 VladimirV.Tkachuk DepartamentodeMatematicas UniversidadAutonomaMetropolitana Iztapalapa,Mexico ISSN0941-3502 ISSN2197-8506 (electronic) ProblemBooksinMathematics ISBN978-3-319-16091-7 ISBN978-3-319-16092-4 (eBook) DOI10.1007/978-3-319-16092-4 LibraryofCongressControlNumber:2015933375 MathematicsSubjectClassification:Primary54C35,Secondary46E10 SpringerChamHeidelbergNewYorkDordrechtLondon ©SpringerInternationalPublishingSwitzerland2015 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper SpringerInternational PublishingAGSwitzerlandispartofSpringerScience+Business Media(www. springer.com) Preface This is the third volume of the series of books of problems in C -theory entitled p “AC -TheoryProblemBook”,i.e.,thisbookisacontinuationofthetwovolumes p subtitledTopologicalandFunctionSpacesandSpecialFeaturesofFunctionSpaces. The series was conceivedas an introductionto C -theorywith the hope that each p volumecouldalsobeusedasareferenceguideforspecialists. Thefirstvolumeprovidesaself-containedintroductiontogeneraltopologyand C -theoryandcontainssomehighlynon-trivialstate-of-the-artresults.Forexample, p Section1.4presentsShapirovsky’stheoremontheexistenceofapoint-countable(cid:2)- baseinanycompactspaceofcountabletightnessandSection1.5bringsthereader to the frontier of the modern knowledge about realcompactness in the context of functionspaces. The second volume covers a wide variety of topics in C -theory and general p topology at the professional level, bringing the reader to the frontiers of modern research.Itpresents,amongotherthings,aself-containedintroductiontoAdvanced Set Theory and Descriptive Set Theory, providing a basis for working with most popularaxiomsindependentofZFC. Thispresentvolumebasicallydealswithcompactnessanditsgeneralizationsin the context of function spaces. It continues dealing with topology and C -theory p ataprofessionallevel.Themainobjectiveistodevelopfromscratchthetheoryof compactspacesmostusedinFunctionalAnalysis,i.e.,Corsoncompacta,Eberlein compacta,andGul’kocompacta. InSection1.1ofChapter1,webuildupthenecessarybackgroundpresentingthe basicresultsonspacesC .X/whenX hasacompact-likeproperty.Inthissection, p thereaderwillfindtheclassicaltheoremofGrothendieck,a verydeeptheoremof Reznichenkoon!-monolithity,underMAC:CH,ofacompactspaceX ifC .X/ p isLindelöf,aswellastheresultsofOkunevandTamanoonnon-productivityofthe LindelöfpropertyinspacesC .X/. p ThemainmaterialofthisvolumeisplacedinSections1.2–1.4ofChapter1.Here we undertake a reasonably complete and up-to-date developmentof the theory of Corson,Gul’ko,andEberleincompacta.Section1.5developsthetheoryofsplittable v vi Preface spacesandgivesfar-reachingapplicationsofextensionoperatorsinbothC -theory p andgeneraltopology. We usealltopologicalmethodsdevelopedin thefirsttwovolumes,sowe refer totheir problemsandsolutionswhennecessary.Of course,the authordidhisbest to keep every solution as independent as possible, so a short argument could be repeatedseveraltimesindifferentplaces. The author wants to emphasize that if a postgraduate student mastered the materialofthefirsttwovolumes,itwillbemorethansufficienttounderstandevery problemandsolutionofthisbook.However,fora concretetopicmuchless might beneeded.Finally,letmeoutlinesomepointswhichshowthepotentialusefulness ofthepresentwork. • theonlybackgroundneededissomeknowledgeofsettheoryandrealnumbers; anyreasonablecourseincalculuscoverseverythingneededtounderstandthis book; • thestudentcanlearnallofgeneraltopologyrequiredwithoutrecurringtoany textbook or papers; the amountof general topology is strictly minimal and is presentedinsuchawaythatthestudentworkswiththespacesC .X/fromthe p verybeginning; • whatissaidinthepreviousparagraphistrueaswellifamathematicianworking outsideoftopology(infunctionalanalysis,forexample)wantstouseresultsor methodsofC -theory;he(orshe)willfindthemeasilyinaconcentratedform p orwithfullproofsifthereissuchaneed; • thematerialwepresenthereisuptodateandbringsthereadertothefrontierof knowledgeinareasonablenumberofimportantareasofC -theory; p • this book seems to be the first self-contained introduction to C -theory. p Although there is an excellent textbook written by Arhangel’skii (1992a), it heavilydependsonthereader’sgoodknowledgeofgeneraltopology. MexicoCity,Mexico VladimirV.Tkachuk Contents 1 BehaviorofCompactnessinFunctionSpaces ............................ 1 1.1 TheSpacesC .X/forCompactandCompact-LikeX................ 3 p 1.2 CorsonCompactSpaces................................................ 13 1.3 MoreofLindelöf˙-Property.Gul’koCompactSpaces.............. 22 1.4 EberleinCompactSpaces............................................... 31 1.5 SpecialEmbeddingsandExtensionOperators ........................ 39 2 SolutionsofProblems001–500............................................. 47 3 BonusResults:SomeHiddenStatements ................................. 459 3.1 StandardSpaces......................................................... 461 3.2 MetrizableSpaces....................................................... 463 3.3 CompactSpacesandTheirGeneralizations ........................... 465 3.4 PropertiesofContinuousMaps......................................... 467 3.5 CoveringProperties,NormalityandOpenFamilies................... 468 3.6 CompletenessandConvergenceProperties............................ 470 3.7 Ordered,Zero-DimensionalandProductSpaces...................... 471 3.8 CardinalInvariantsandSetTheory .................................... 472 4 OpenProblems .............................................................. 473 4.1 SokolovSpacesandCorsonCompactSpaces......................... 475 4.2 Gul’koCompactSpaces ................................................ 477 4.3 EberleinCompactSpaces............................................... 478 4.4 TheLindelöf˙-PropertyinC .X/ .................................... 480 p 4.5 TheLindelöfPropertyinXandC .X/ ................................ 482 p 4.6 ExtralandExtendialSpaces............................................ 485 4.7 Point-FiniteCellularityandCalibers................................... 488 vii viii Contents 4.8 GrothendieckSpaces.................................................... 489 4.9 Raznoie(UnclassifiedQuestions)...................................... 491 Bibliography...................................................................... 493 ListofSpecialSymbols.......................................................... 513 Index............................................................................... 519 Detailed Summary of Exercises 1.1.TheSpacesC .X/forCompactandCompact-likeX. p Somepropertiesofzero-dimensionalspaces .................Problems001–005. Densealgebrasinfunctionspaces ..........................Problems006–007. Stone–WeierstrasstheoremintheformofKakutani ................Problem008. Uniformlydensesubspacesoffunctionspaces ...............Problems009–010. Somepropertiesofk-directedandk-perfectclasses. .........Problems011–013. ThesubsetsrepresentativeforC .X/ ............................Problem014. p ExpressingpropertiesofC .X!/intermsofC .X/ .........Problems015–018. p p ObtainingC .X/2P whenP isan!-perfectclass. ..............Problem019. p Lindelöf˙-propertyinC .X/andC .X!/. ................Problems020–025. p p NormalityinC .X/foraLindelöf˙-spaceX. ..............Problems026–028. p Normalityisdensesubsetsofproducts ...........................Problem029. LindelöfpropertyofC .X/forcompactspacesX ...........Problems030–032. p CompactEG-spacesareFréchet–Urysohn ........................Problem033. AnymetrizablespaceisEG .....................................Problem034. EmbeddingspacesinC .X/foracompact-likeX. ..........Problems035–043. p Closuresofcompact-likesetsinC .X/forcompactX. ......Problems044–049. p HurewiczspacesandtheirC ’s ............................Problems050–059. p EmbeddingC .X/insequentialspaces .....................Problems060–063. p Radialityandpseudoradialityinfunctionspaces. .............Problems064–075. LindelöfpropertyinC .X;D/anditsderivatives ............Problems076–078. p Nowheredenseclosedsetsincompactspaces .....................Problem079. NormalityofC .X/forcompactX underMAC:CH ........Problems080–081. p LindelöfsubspacesinC .X/foracompactLOTSX. .............Problem082. p ClosuresofLindelöfsetsinC .X/foracompactX. .........Problems083–084. p OnsubsetsofC .(cid:3)C1/separatingpointsof(cid:3)C1 ...............Problem085. p LindelöfsubspacesaresmallinC .X/foradyadicX. .......Problems086–087. p CompactainC .X/forLindelöfX underPFA ..............Problems088–089. p Lindelöfpropertyofcountablepowers ......................Problems090–091. ix
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