作者	J.B.Conway
出版社
出版时间	1997-09-01

特色：

片断：Chapter13ReturntoBasicsInthischapterafewresultsofasomewhatelementarynaturearecollected.Thesewillbeusedquiteoftenintheremainderofthisvolume.?RegionsandCurvesInthisfirstsectionafewdefinitionsandfactsaboutregionsandcurvesintheplanearegiven.Someofthesemaybefamiliartothereader.Indeedsomewillberecollectionsfromthefirstvolume.Beginbyrecallingthataregionisanopenconnectedsetandasimplyconnectedregionisoneforwhicheveryclosedcurvpiscontractibletoapoint(see4.6.14).InTheorem8.2.2numerousstatementsequivalenttosimpleconnectednessweregiven.Webeginbyrecallingoneoftheseequiv-alentstatementsandgivinganother.DonotforgetthatCoodenotestheextendedcomplexnunibersandGdenoteStheboundaryofthesetGin旵.Thatis.whenGisboundedandwhenGisunbounded.Itisoftenconvementtogiveresultsaboutsubsetsofthecxtendedplaneratherthanabo'utC.IfsomethingwasprovedinthefirstvolumeforasubsetofC,butitholdsforsubsetsofCwithlittlechangeintheproof,wewillnothesitatetoquotetheappropriatereferencefromthefirsttwelvechaptersasthoughtheresultforCwasprovedthere.1.1Proposition.IfGisaregioninC,thefollowingstatementsareequivalent.(a)Gissimplyconnected.(b)C\Gisconnected(c)Gisconnected.Proof.Theeauivalenceof(a)and(b)hasalreadybeenestablishedin(8.2.2).Infact,theequivalenceof(a)and(b)wasestablishedwithoutassumingthatGisconnected.Thatis,itwasonlyassumedthatGwasasimplyconnectedopenset;anopensetwitheverycomponentsimplyconnected.Thereadermustalsopayattentiontothefactthatthecon-nectednessofGwillnotbeusedwhenitisshownthat(c)implies(b).Thiswillbeusedwhenitisshownthat(b)implies(c).