Continuous extension in topological digital spaces

CONTINUOUSEXTENSIONINTOPOLOGICALDIGITALSPACESERIKMELINAbstract.Wegivenecessaryandsufficientconditionsfortheexistenceofacontinuousextensionfromasmallest-neighborhoodspace(Alexan-drovspace)XtotheKhalimskyline.Usingthisresult,weclassifythesubsetsA⊂XsuchthateverycontinuousfunctionA→Zcanbeex-tendedtoallofX.WealsoconsiderthemoregeneralcaseofmappingsX→Ybetweensmallest-neighborhoodspaces,andproveadigitalver-sionoftheno-retractiontheorem.1.IntroductionTheclassicalTietzeextensiontheoremstatesthatifXisanormaltopo-logicalspaceandAisaclosedsubsetofX,thenanycontinuousmapfromAintotheclosedinterval[a,b]canbeextendedtoacontinuousfunctiononallofXinto[a,b].Indigitalgeometryitismorenaturaltostudyfunctionsfromadigitalspace(definedbelow)totheintegers.Havingequippedthespaceswithsuitabletopologies,wemayconsidercontinuousfunctionsandthecorrespondingextensionproblem.InthispapertheproblemissolvedforafairlygeneralclassoftopologicaldigitalspaceswhentheintegerlineisequippedwiththeKhalimskytopology.Theresultisageneralizationofapreviouswork[Me03],wherethedigitalspacesconsideredwererestrictedtoZnequippedwiththeKhalimskytopology.Herman[He98]givesthefollowinggeneralgraph-theoreticaldefinitionofadigitalspace:adigitalspaceisanon-emptysetVequippedwithabinary,symmetricrelationπ,calledtheadjacencyrelation,suchthatVisconnectedunderπ.ThismeansthatgiventwopointsxandyinV,thereshouldexistafinitesequencex=a0,a1,...,an=yofpointsinVsuchthat(ai,ai+1)∈πforeveryi=0,...,i−1.Thisdefinitionisindeedverygeneral;Visallowedtobeasetwithoutanygeometricalrestriction.Forabasicexample,thinkofEuclideanspaceRn,Vasanarbitrary,butfixed,setofgridpoints(forexampleZn)andπasarelation,tellingwhichofthesepointsareneighbors.Inthispaperwewillconsiderdigitalspacesthatarealsotopologicalspacesandwherethetopologicalnotionofconnectednessagreeswiththenotionofconnectednessindigitalspaces.Moredetailsonthetheoryofsuchspacesandapplicationsinimageanalysiscanbefoundinforexample[Kv89,KR89,KKM91,Ki02,EL03].2000MathematicsSubjectClassification.Primary54C20;Secondary54C05,54F07.Keywordsandphrases.digitalgeometry,Khalimskytopology,Alexandrovspace.12ERIKMELIN2.BackgroundInthissectionwereviewsomedefinitionsandresultsthatwillbeusedinthispaper.2.1.Topologyandsmallest-neighborhoodspaces.Inanytopologicalspace,afiniteintersectionofopensetsisopen,whereasthestrongerre-quirementthatanarbitraryintersectionofopensetsisopen,isnotsatisfiedingeneral.Intheclassicalarticle[Al37],Alexandrovconsiderstopologicalspacesthatfulfillthestrongerrequirement,wherearbitraryintersectionsofopensetsareopen.Inhispaper,AlexandrovcalledsuchspacesDiskreteR¨aume(discretespaces).Unfortunately,thisterminologyisnotpossibletodaysincethetermdiscretetopologyisoccupiedbythetopologywhereeverysetisopen.Instead,followingKiselman[Ki02],wewillcallsuchspacessmallest-neighborhoodspaces.AnothernamethathasbeenusedisAlexandrovspaces.LetNX(x)denotetheintersectionofallneighborhoodsofapointxinatopologicalspaceX.Ifthereisnodangerofambiguity,wewilljustwriteN(x).Inasmallest-neighborhoodspace,N(x)isalwaysopenandthusaneighborhoodofx;clearlyN(x)isthesmallestneighborhoodofx.Wemaynotethatx∈N(y)ifandonlyify∈{x},whereAdenotestheclosureofthesetA.Conversely,theexistenceofasmallestneighborhoodaroundeverypointimpliesthatanarbitraryintersectionofopensetsisopen;hencethisexistencecouldhavebeenusedasanalternativedefinitionofasmallest-neighborhoodspace.AtopologicalspaceXiscalledconnectediftheonlysetswhicharebothclosedandopenaretheemptysetandXitself.NotethatN(x)consideredasasubspaceofXisalwaysconnectedandthatatwo-pointset{x,y}isconnectedifandonlyifx∈N(y)ory∈N(x).Apointxiscalledopeniftheset{x}isopen,andiscalledclosedif{x}isclosed.Ifapointxiseitheropenorcloseditiscalledpure,otherwiseitiscalledmixed.Kolmogorov’sseparationaxiom,alsocalledtheT0axiom,statesthatgiventwodistinctpointsxandy,thereisanopensetcontainingoneofthembutnottheother.AnequivalentformulationisthatN(x)=N(y)impliesx=yforeveryxandy.Thisaxiomisquitenaturaltoimpose;ifxandyhavethesameneighborhood,thentheyareindistinguishablefromatopologicalpointofviewandshouldperhapsbeidentified.TheseptationaxiomT1,ontheotherhand,istoostrong.Itstatesthatpointsareclosedandinasmallest-neighborhoodspacethismeansthateverysetisclosed.Henceonlyspaceswiththediscretetopologyaresmallest-neighborhoodspacessatisfyingtheT1axiom.Remark1.Thereisacorrespondencebetweensmallest-neighborhoodspacesandpartiallyorderedsets;namelyifwedefinex4yifandonlyify∈N(x).Thisrelationisalwaysreflexiveandtransitiveandisanti-symmetricifandonlythespaceisT0.Thentheorderisapartialorderandiscalledthespecializationorder.ItwasintroducedbyAlexandrov[Al37].Itisnothardtoseethatafunctioniscontinuousifandonlyifitisincreasingforthespecializationorder.Aconsequenceisthatourresultscanbeformulatedinthelanguageofpartiallyorderedsetsinsteadoftopologies,ifoneprefers.CONTINUOUSEXTENSIONINTOPOLOGICALDIGITALSPACES3Figure1.ConstructionoftheKhalimskyline.2.2.Topologicaldigitalspaces.GivenatopologicalspaceXwemaytrytoidentifyitwithadigitalspaceinthesenseofHerman.Itisnaturaltodefinethebinaryadjacencyrelationπby