Atlantis Studies in Mathematics Series Editor: J. van Mill Michael G. Charalambous Dimension Theory A Selection of Theorems and Counterexamples Atlantis Studies in Mathematics Volume 7 SeriesEditor JanvanMill,UniversityofAmsterdam,Amsterdam,TheNetherlands AimsandScope Withthisbookseries,weaimtopublishmonographsofhighqualityinallareasof mathematics. Both research monographs and books of an expository nature are welcome.Thisseriesisthe continuationofthe “MathematicsStudies”,previously published by Elsevier. All books published after November 2010 are promoted, distributedand sold by Springer,bothas e-booksand in print. The booksare also partof SpringerLinkand includedin the relevantSpringersubjectcollections.All bookproposalssubmittedtothisserieswillbereviewedbytheSeriesEditor.After the manuscript has been completed, it will be entirely reviewed by one of our editorsorreviewers.Onlyafterthisreviewwillthebookbepublished. Moreinformationaboutthisseriesathttp://www.springer.com/series/10070 Michael G. Charalambous Dimension Theory A Selection of Theorems and Counterexamples 123 MichaelG.Charalambous DepartmentofMathematics UniversityoftheAegean Karlovassi,Samos,Greece ISSN1875-7634 ISSN2215-1885 (electronic) AtlantisStudiesinMathematics ISBN978-3-030-22231-4 ISBN978-3-030-22232-1 (eBook) https://doi.org/10.1007/978-3-030-22232-1 MathematicsSubjectClassification(2010):54F45,54E35,54Exx,54-XX ©SpringerNatureSwitzerlandAG2019 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. 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Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland ForIrene Preface Dimensiontheoryisabranchoftopologythatconcernsitselfwiththepropertiesof certain functionsd that assign to a topologicalspace X a memberd(X) of the set {−1,0,1,2,...,∞}sothatd(X) = −1ifandonlyifX istheemptyset,d(X) = d(Y)ifXandY arehomeomorphicandd(X)=nifXistheEuclideann-spaceRn. Suchfunctionsarecalleddimensionfunctions. The subject grew out of efforts by mathematicians at the beginning of the twentieth century to give precise meaning to the naive notion of dimension and, in particular, to prove that Rm and Rn are topologically equivalent only when m = n. That Rm and Rn are not homeomorphic for m (cid:3)= n was first proved by Brouwer in a paper dated 1911. Soon afterwards, he gave the first definition ofadimensionfunction.Brouwer’sdimensionfunction,calledDimensionsgrad,is of no particularimportancein moderndimensiontheory.The principaldimension functions in order of significance are the covering dimension, dim; the large inductive dimension, Ind; and the small inductive dimension, ind. In the early 1930s, modifying the definition of Brouwer’s Dimensionsgrad, Cˇech defined Ind and developed a theory for perfectly normal spaces. He also defined covering dimension, basing his definition on certain properties of covers of the Euclidean n-cubeIn observedbyLebesgueina1911paper.Butthefirsttheoryofdimension thatwasdevelopedisthatforind,whichwasfirstdefinedandstudiedindependently byUrysohnandMengerintheearly1920s. BythebeginningofWorldWarII,thetheoryofthedimensionofseparablemetric spaceswasvirtuallycomplete,themostsignificantresultbeingthecoincidenceof the three-dimensional functions. The essential parts of the theory were presented in the classic book of Hurewicz and Wallman, Dimension Theory, published in 1941. After World War II, research focused on extending the results to more general spaces. The greatest achievement of this period is considered to be the creation of a satisfactory theory for metric spaces, whose most important result is the coincidence of dim with Ind, proved by Kateˇtov and Morita independently in the early 1950s. However, the efforts to extend many results to more general vii viii Preface spaceswerefruitfulmainlyinanegativemanner.Typically,severalopenproblems remainedunansweredfordecades,onlytobefinallysettledbyacounterexample. Thedimensiontheoryofgeneralspaceshasbeenpresentedinseveralexcellent books.Iamparticularlyindebtedtothefollowingthree: (a) A.R.Pears,DimensionTheoryofGeneralSpaces,CambridgeUniversityPress, Cambridge,1975 (b) R.Engelking, GeneralTopology,HeldermannVerlag,Berlin,1989 (c) R. Engelking, Theory of Dimensions, Finite and Infinite, Heldermann Verlag, Lemgo,1995 The main purpose of this book is to present a selection of some of the most interesting of the relatively recent counterexamples in the dimension theory of general topological spaces. Regrettably, many important but technical examples havenotbeenincluded.Transfinitedimension,constructionsthatmakeuseofset- theoretical assumptions beyond ZFC and examples of compact Hausdorff spaces with distinct small and large inductivedimensions are some of the topics that are not discussed here. Generally, the emphasis is on topics that can interest a wide range of potential readers and are not covered in existing books. Well, over half of the contents of this monograph do not appear in other books on dimension theory.Thenewmaterialincludesthreegeneralmethodsofconstructingtopological spaces: Mrówka’s ψ-spaces, van Douwen’s technique for constructing examples andFedorcˇuk’sresolutions. Althoughthe bookis not intendedto serve as a comprehensivereferencework ondimensiontheory,itcoverssufficientmaterialforagraduatecourseandcanalso beusedforself-study.Whilefamiliaritywiththestandardfactsofgeneraltopology isassumed,everyefforthasbeenmadetomakethebookself-containedandreader- friendly.Alltopologicaltermsarerecalled,andtheirdefinitioncaneasilybelocated withthehelpoftheindex.Thebookcontainsalargenumberofexerciseswithhints for a proof of most topological results that are used in the book without having beenproved.Thereisanequallylargenumberofexercisesthataremeanttotestthe reader’s understandingof the material or serve the purpose of introducing further resultsthatcanbeestablishedwiththehelpoftheoremspresentedinthebook. There are brief notes at the end of almost every chapter where the authors of mostresultsappearinginthebookarecited.Furtherinformation,especiallyonolder results,canbefoundinthethreebooksmentionedabove,whichcontainawealthof historicalandbibliographicalnotes.Thereadershouldbeawarethatthe proofsof manyresultsinthebookarenotthesameastheoriginalproofs. I thank Jan van Mill for suggesting the idea of writing a book on dimension theory. I also thank Dimitris Georgiou for pointing out several misprints, inaccu- racies and other errors in the text. I am grateful to Jerzy Krzempek, who read a large portion of the book with a critical eye and made extensive suggestions thatled to considerableimprovements.Of course,I take fullresponsibilityforthe imperfectionsthatremain. Nicosia,Cyprus MichaelG.Charalambous Contents 1 TopologicalSpaces.......................................................... 1 2 TheThreeMainDimensionFunctions................................... 7 3 TheCountableSumTheoremforCoveringDimension................ 15 4 UrysohnInequalities ....................................................... 23 5 TheDimensionofEuclideanSpaces...................................... 27 6 ConnectedComponentsandDimension ................................. 31 7 FactorizationandCompactificationTheoremsforSeparable MetricSpaces ............................................................... 37 8 Coincidence, Product and Decomposition Theorems forSeparableMetricSpaces............................................... 41 9 UniversalSpacesforSeparableMetricSpacesofDimension atMostn..................................................................... 45 10 AxiomaticCharacterizationoftheDimensionofSeparable MetricSpaces ............................................................... 51 11 CozeroSetsandCoveringDimensiondim .............................. 61 0 12 ψ-SpacesandtheFailureoftheSumandSubsetTheorems fordim ...................................................................... 75 0 13 TheInductiveDimensionInd ............................................ 85 0 14 TwoClassicalExamples.................................................... 99 15 TheGapBetweentheCoveringandtheInductiveDimensions ofCompactHausdorffSpaces............................................. 107 16 InverseLimitsandN-CompactSpaces .................................. 115 17 SomeStandardResultsConcerningMetricSpaces..................... 129 ix x Contents 18 The Mardešic´ FactorizationTheoremand the Dimension ofMetrizableSpaces ....................................................... 139 19 AMetrizableSpacewithUnequalInductiveDimensions.............. 147 20 NoFiniteSumTheoremfortheSmallInductiveDimension ofMetrizableSpaces ....................................................... 153 21 FailureoftheSubsetTheoremforHereditarilyNormalSpaces ...... 155 22 AZero-Dimensional,HereditarilyNormalandLindelöfSpace ContainingSubspacesofArbitrarilyLargeDimension................ 165 23 CosmicSpacesandDimension............................................ 171 24 n-CardinalityandBernsteinSets......................................... 183 25 ThevanDouwenTechniqueforConstructingCounterexamples ..... 187 26 No Compactification Theorem for the Small Inductive DimensionofPerfectlyNormalSpaces................................... 201 27 NormalProductsandDimension......................................... 205 28 FullyClosedandRing-LikeMaps........................................ 213 29 Fedorcˇuk’sResolutions .................................................... 223 30 CompactSpacesWithoutIntermediateDimensions.................... 231 31 MoreContinuawithDistinctCoveringandInductiveDimensions... 235 32 TheGapsBetweentheDimensionsofNormalHausdorffSpaces..... 245 Bibliography...................................................................... 251 Index............................................................................... 259