锥度量空间中的一些新的拓扑性质.pdf

数 学 杂 志 Vo1．35(2015) NO．3 SoM E NEW ToPoLoGICAL PRoPERTIES IN CoNE M ETRIC SPACES HUANG Hua-ping，XU Shao—yuan (SchoolofMathematicsandStatistics，HubeiNormalUniversity，Huangshi435002，China) Abstract： In thispaperjwe study somepropertieswith respectto conesin conemetric spaces．Byusingthenotionsofcompleteness，weobtainnestedclosed—balltheorem insuchspaces， which improvessomepreviousconclusionsinmetricspaces． K eywords： normalcone；conemetricspace；nestedclosed—balltheorem 2010MR SubjectClassification： 47H07；46A55 Documentcode： A ArticleID： 0255—7797(2015)03-051306 1 Introduction Nonlinearfunctionalanalysis，especiallyorderednormedspaces，hadsomeapplications inoptimizationtheory[1】．Inthesecasesapartialordering “ ”bymeansofwhichcertain elementscan becomparedbetterthan crudeestimatesin termsofanorm ，isintroduced byusingvectorspacecones．In2007，HuangandZhang[3]definedtheconemetricspaces with adifferentview． Infact，they substitutedanormedspaceinstead oftherealline， butwentfurther，definingconvergentandCauchysequencesinthetermsofinteriorpoints oftheunderlyingcone． Sincethen，many scholarsfocusedon the investigationsin such spaces[4-5]．Inrecentyears，sometopologicalpropertiesinconemetricspacesbecamethe centerofstrongresearchactivities[6-8]．Throughoutthispaper，wegivesomeproperties oncones．Furthermore，wepresentnestedclosed—balltheorem in conemetricspaces．All resultsdirectlygeneralizeandreplenish someassertionsinmetricspacesandsomeprevious resultsinconemetricspaces． Weneedthefollowingdefinitionsandresults，consistentwith[3]，intheseque1． LetE bearealBanach spacesandP asubsetofE ．By0wedenotethezeroelement OfE andby intJF)theinteriorofP．ThesubsetP iscalledaconeif： (i)Pisclosed，nonempty，andP≠{)； (ii)a，b∈ ，a，b 0， ，Y∈P ax+by∈P； (iii)Pn(一P)={)． Recei