Introduction to

Topolo9Y

Introduction toTO POLOQY

10 . r . 6opKc o lJlt'f . H . M . E /nt 3111llJ:OIl , JI . A . H 3pllHlIe DHOf.

T . H . IoMeHkO

BBELlEHHE B TOOm a n-tlO

JiJ.a a Te..'1bC'T BO B .. ICWWII W1ICoJlaM """",

Introduction to

lf~p~~~~yYU. BORISQVICH.N. BUZNYAKOV.

VA, IZRA ILEVICH.

T. FOMENKO

Translated rrom the Russianby Olea Efunov

MIR PUBLISHERS MOSCOW

First pu btl. hed n u

~ from the 19Itl It...,.;.,. edHloa

o "m~ ..8~ mKQM)O, 191() EnalW> u ana1f,Ooa. "'Ill PIIblWlt rs 15lll'

,J'lRST NOTlOm OF TOPOl.OCiY

1 .W!laI Il~1. ~1Wotioft of ,be COflCqltJ of ~I*'C IIIld '''''''ion3. From I rIlftric to lQPOlosical watt4. The notion of ItiaDaan _fad:5. SomnbiD& about knot,

FlIfIher; .-lin,

""",.

""

GENSRAL TOPOLOGY

l. Topolotleal lpoI

3. Smocxh IlWIifolds 16 .C. Smoofh fu:nctionllin. manifold and IlIIOOlh parthian of unil, 173,. MaPl' inas of manifolds ISO6. Tan&enl bUndle lind l&t\ICnlia l map 1831. T;onaent V(Q'r u di fferential operator . D1rr~ll.1 or l\.Iocllon Ind cotaJ\&enl

bundle 199I. Vector rlddl on JIlIOOCtI JIl&Ilifoids 2089. Fibre bund le...nd --mnp 113

10 .sr-Il fltDClioD on rDallifdd &.s ccUular Slnoc:twc of ........fold (na>Dllk) 215I I . ""....'S ......l'llle cmieaJ poUilllDd its mda 14011. D=:ribi"l'--DPJ Inx 01 mani fold bJ _ of critlc:al ....115 244

Funis" rndina 249

,HOMOLOGY THEORY

I . PJd lmina l)l notC!l1. HomolotY srGI'PS of dWn cucn~, . Homolot.Y.p CNpl of limplicial eomplaa4. Sift&ulaT lIomoiosJ lkcory, . HOlnOIDsY lMory a,xiomI:6. HOIIIOIoIY ,"",ps of~ Devft of mappina7. Homolou ITClI.1J* of cdl COInJIk:HsC. Ellkr charaderlstic and urodld~ nllmbn

Further ladln.

lIl",nrll.llonliw._......-.

SKbjtct Indelt

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PREFACE

TopoiolD' is a subject t~t has only~nll' bem inuod Ud into the:c:urrituJumo f mallianatic$ departmmts. Howc'!er , it does , in w r op inion, play quile a considuab4c: ro le: with respc:a 10 unl venlty math..matics ed ualtlon as III. whole:. It isbard )' possI ble: to design courses In math emat ical anaIy.loi" differen tial equation"diffCTC'ntllll geometry , medlanks, and funetilmal analy sis that H'relpond to themode m $late o f these dJ5d pllnes without involvIng tOpOlogical concepts. It "therefore cuentiltJ to ac:.quaint atuden ts willi topoloaJcal methOcordaJIce: .nth both the autbors' prefere_ and lbdr apc::ric:ncc:as Icc:rwCQ and rac::arehc:n . Jt dull wit.b those area of topoJosy thal we I'l:>OStcJosdy rdated to fundamc:n~ COWKS in smual matbetnaticJ:and applieadOlU. Thematmallnves .aICClIIrft" lffe choice as In bow he Of she mil' want to drsian hi, orher own to pology course and lleminar cla.s.sa .

We draw your atten tion to nwnbu of dcvka we have: l.lSC'd in lhis book inorner to Introduce: gentrlll tOpOlOlD' railer. We ha.-e Iher ..Core:introduced conllrue-tive conc:c:pl.S , fo r c:urnple. those related to the:noI lon o f faetor ' pace, m uch earlierth M the ether notlolll of general to pe loSY. Th il ma kes il posdble for studenu tostudy ImpOnant exam ples of mani folds (two-d immlional surfaces, f)rojealvespaces. orbil spaces, et c.) as topolQlkalsp&CCS. Later (01. IV) ,mooth 5lnlC\utnare der".c:d on them. The th COl}' of two-d imen sional sW"f_ b not ClOfInnc:dee e neplaee: but 11 distributed IIIIKIIlJ$I. Ch . I , Cb . II and Ch . JlI IS and when the bas icideas o f amcallopo&osy are devdopc:d. The JIOlion$ o f ealqory and functor an:Inlrodueed inlO hOlllOtopy lhcoty quite early; likewise, the Idea ollH a1acbnl iu-t ioo of lopoJnlic:a1pt'Ob~ The functori.al apprOlldt bdps lIS apound holnotopyU1d homoJoay theories uniforcnJy and compktc: tbe dcsaiptioo of variousboIDo kl$)' theories with the Stc:e:n rod -Eilm bc:rl a.Uom.alks,lIIUiD, up , to limit a -ImI, lor the abfeneeo f the:proofof W invariana: of limptil:ial bomolosy theory Inthis IW booJc. Moreover. thehornotopy almpul iUilnll edmlque (Cb_ JU) if reduced10 the c:alcubtion of the fundammt.al arou p! of the a raunfc:n::-.t an d ckJse:dMlr

fKC$. "The eq uality,.... (5" ; Z ) "'" Z . II .. 2 u, ho>ftvu, liven '/Ii'ithoul proo f andsaves as a basis for me IntrodlK1ioo o f the dqrcc of. nuoPPinl of sphcns an d thecharacteristic: of vector r.eld (willi the Brou wer and fundamental theor em ofalarM' bcinl Qedueed); while in thc bo mo locYIlfOUP section (Cb. V). the techniq ueb U1 cl:Kkd to CU(t SC'QUCIJCaI. In p*rtil;Ular , the lJou p H,. (5" : Z) is com puted, andthe Brollwcr and Lefschcu. rl!led-poi nt theorems arc pro~. Tn spite of havln!prepared cY

First Notionsof Topol09Y

The purpose of this chapter is 10 pupare the reader for thesyllemalic: stud y of lopoloS)' as it b expounded in the slibseqUentchapleTs. Ou r PUl'pQ5C here is to review the prob lems , whose sc tu-lion has led 10 the formation of lopolol ' as a mathematicaldi scipline an d to its devdop~nl at present . We il1sodUcuss th ebqinnings of lbe notions o f topological space an d manifold .

I. WHAT lS T OPOL OGY?

Quant 1 moi~ toute, Ie' vole'di_ 0n

"lnllodUClion '0 Topolo&y

anothl:l". Thu.s, the qualitative properties of the sphere are rhcse whidt II shares with..u its homeomorJIhlc repDeRs, or. in olher words, those which au prcseO'ed unl1CTIIomeomorphl$lll$.

It Is evident that hornromorphl$ms and the q ualitative properties ct otherligula may be lIisRl50Kd as well. It is a1Io conventional to call the qualitative prop-erti es fopolOJimJpropn1;u. In the aboy!! example. one of the topoloaiea! prcper-lin of lhe sphen: is obviotu. I.e., Its Intearlty (or con ncetcdne$3) . II.! more subtleproperties art ~vealed if an attunpt is made to establish a homeolriolllhism o f thesphere, IllY, with th e bal l. It is CASY to co ncllKlc that such IIhomeomorphism is im-pm$ible. However. in orde r 10 prove lbat . il ill necessary 10 show the variouslopo lo&leal properties of spheru and ball" . One of these is the 'con tract ibillly' o fthe ball ln!o one of its lXIinu by changing it smoothly. t.e..eonuaelina it lI1Qn, iuradlllOWltdS lbc eeeee, and the 'non-

O . I. F",rs' Notions o f Topolosy u

I. . A. L1lStem ik. L. O . Schnirt'lman . M . MOTSe . A. N. Tihonov. L. S. Po.!-tryqm. A.. N. Kolmoaorov. E. O:dt . et aI . Sovla malhmtal M:Wis have made profou nd an d u len slve contrib utio n to the de vdopment o f topolOIY u a whote.

To d escribe preciJely the results oblain~ (and eveu to pose problems) II impos_dble wllhollt being acquainted wilh the e1ementl ot general and lI1aebraie topololY.Heee, we Jive merel y Hlfl\e idea of the p roblems that have sdmulat~ topol ogKal,""""'-

If s' is clreumfcrcnet: Ott the Euclidean pl.Me: R 1 then the !d R 1 '\. S I dOm'pt:pes into two muh llllly co mpkmoeutary open scu . vU. the Interior A and ahc ex-lerior 8 ofs'. The cil'CUlllfe1CftCl' s' serva u a~r betli'em A and B . Can asiDspk: contlnuous path be drawn from an arbitrary point II ti A to til arbitrarypoint b e 8 $0 that it docs not int ersect the sep'n.tor 5 " (A.simp ff' mli"IlDIUINIIII is a homromOl'Jlhk: mappilll o r the lineseament 10 . I ) of the number llne intolhe pbne.) The M swn is negative. In tee. if p lx. y) is the EIlt:Ud ean d islanee be-tween PQlnu)t , y of the pbne R 1 and "I'(t) is w eh a path, 0 " t" t , "1' (0)

.. Inll od ..etioa 10 TopoJoty

analysis, the qvestion wha bet"Dr not there aisl sol uliorl5 of eq uat ions o f th e f()l'1llI~) .. O. ( I)

(2)! (.l")+111 cx.men:/(.l") is.~ial or IJM)fC~mpliQted fUlK1ion . Equa tion (I) b equivalent10 the equ.aaion

o r. when F~) .. l (x ) + III, 10 the equationF(x} d x. (3)

The &olu tions o f eq uation (3) arc ca1Icd tbe:~~ poinu 01 1M nulppfn , F. Ifequlllio o (I) i' eaor. i .e. ,lf iC is . system o f ~uat;ons in seven! unknowl. lhcn theequivlakat equatiol'l (l) is llbo vector and , tba rIon. the fl.UCl poinlJ;1i.e Ia . many-

dlm~na1 Euc:lideaD space.An exuemdy iJI1ponaat taN; is to find sum~icnl'" . m etal and ef lectin lew.

that wl1I illdicale if flMd poinu cma.~ obtained. rcmacbtMe result thai Ia4vcry eMcrWYC ap plications ift modem researc:h . It is IUfpri$iqly simple 10 lormulMc: any c:onlll'l\lOus mapplnJ of. bounde d . con vu, d OleCl let inlo il.5t:lf has .fixed point. Cc a vex $etS may be considcd bo rn in tbe lh r~dimcnsion.l andmany-dimenloional Euclidean Ipace. Fo r example. Ii coniinu oul mapping inlo ittd fof. cloud (I.e . COMldcred alOI\I with its' bou nd ary) disc In p lane or bah in space necessaril y 1Iq. fIXed point .X~ 2- . Show IbM an anaJoaue of the Brouwu theQftm for aD annulus dOC$ not..... ....

The Bto _ tbeomn _ clevdopcd by H. Hop f, S. l d sdletz et aI . It Willabo ,cnCBliud fo r lM: mappiDp o f fWln ion spacrs OCcllol, BirkhoIf, Scha ud er,Way) which u lmded its Ippaieatiolls. II should be DOI.ed thal e'ftO H . Poinean:himsdf_ intcresced in the ~lWtcnc:c thlrenu for ( txed poin u when [eduQnl eer-tlin problems In cdaWtl mccblnia to Ih~m .

w~ emphlsae thaI the tluee probkms de&

ClI. I. Flnt ND< loou: of Topololr"

cumulal N due to the ru.eareh of such mathem&1kians . s Hopf. POIlI~acin .Whitney . Stnrod. Eilenbug. M:oc::Lan e. Wh itehead eI a l. II Ihen bcc.ameDesury 10 WOC"k out. uni fied approach 10 all the various datalhal. have bcnl ob-IlIiMd and 10 create new general rnetbo

"In l, od uGlio n to Topol"6)'

The functiolllp from EKerdscs l and 1 are natur ally call ed the dlslanc:n be l WffIlIhl! t!umenrs o f the COrresponding seu.

To introduce a general ecncepr of d1uance, recall the definition of the productof lwo " IS. If X and Y lLh: two seu then the ir produ ct X )( Yis lhe $d. consisting of..... ordered pairs ~. y), where x E X, Y E Y. In particular, the product X x X isdcrmcd .

DEFINITION I . A set Kalong willi the llUIpping /:I: X x X - R 1(iQto th e numberaxis ), llliSOCiat;Dg each pai r ~,y) Ii! X x X wit h II real number p~. y) and iJltis f yiogProperties I-I V , is called IImetricspo~ and denoted by (X. pl.

Thc ffiappilliP iscalkd thc disuma or m etric 011 thcspgX. The clemfllS of Xarc usually caUed points.

Any ser may be made into II metric space by cndowina it wilh the metricdescribed in Exen:isc ' '' , Such a met ri0 0 is an intege r and, when decomposed intoprime fKton, co ntains a power P"'. then we pill \Op(n ) CI a . EJllend the function \Opfro m the set o f positive integers 10 the SCI Q '\. 0 of rational number$ without zerousing llle formula vp(lI:r fs) '" vp(') - vp(s). PUI

PVc , )')= p - -" fJ< -Yl , x*y.pVc , x ) = 0

0., I . Fiul N O. Ihere iI a naturalnumbcTlIo{d web lUI p. (.t~. II) a:CIO II' ll'erNly speak of lhe eonvCJlence ofthb; sequence10 an e{emeDt xo = xo{r): x~ !. XO' Sucha eonvtraenu is often Aid to be IIffi/rNm0/1 tIN Kfnwllf 10. I IExzrtiu 7 . Show that the ieCI~lZ of ruon ions X.(I) - 112re-'" on the sq;mento " l " 1 eonversa to the aere funetiOllfor any I but doa nol converae llIIifOf1ll-

".We lKlW ddittethc notion of continuous mappilla of a metric space tx. pJwo. 1lloCtrlc space (Y,1',.OEFlNITION 2. Ld/: X _ y~ . mappina oh SC1 X illtO a lei: Y. If, roe lily point

Jto Ii X and any ieClucna x.. ~ Xo in X. the 5l!qucncc o f the im.a&eI in Y con..efll'S10/(.t,};ff,1c~) ~ f(r,}. chen the mappi ns / b called . l'(IIIfill llOUS tnQPP;"1 oj theme/ri" spIKe (X, Pt) into thr metric: space (Y. po) .

Thil definition II C"ildenlly a aeneraHUt lon of the concept of continuouJnumerical funnion; It (oven . gfeat dell!of mappinas of l eometrk figures in Euclidean spaces.

If the prope rty of cont inuity given by Definition 2 is eonsldeeed at a ernainpoint Xo- then . definition of a continuous mappinl ll the point Xo it obt.lned.Eureka.r . Let S2 be a sp!lrre in the Euclidean Ipa~e R ) with its cenue . Ihe origin. PUltinajfJt) - - x (II is paine symmetry). pn).-e that f ic ~Onl inlMlllS .9" . Givt an CQlJlple of a cont inuo~ rmtppin& of a planuquare into itsdf tlW has(IUd poinn only 0.0 the boundary.

Ob..iousiy. an equlvaJmt dc(lIIition of a contimlOlU mappina o r metril:~ma)' also be p ..en in tmns o r e, f .

"mlIfJPbr,j: X - Y iJCOIIt/II_ lf fOf any xoe X alld fOl" InYt :> 0, lhar is" ~ ,(t,:rt,} > 0 suc:b ttw P)(f(.t). f lit,}) -c e as soon as p, (r. xli < 6.

"InUod"elio D'

Ch. I. Am NocionJ of TQ4lOloIy

t_n----------_n-n ---! dcdasaauv'ilinearlntqra!inthee.pIlneCalonl acenllinpf;th1 " lU).O " s " I .iOin1nl llte pOmU 10 0IJld z. where fi. il one of Ihc brlDdles of IIIe: many....alucd aJaebrale lUrK.lion .I ven by lhc equalion .. ' - < O. The """'" int.gral can a1Io be considered as the cur_vilinear intqral

j R(r)d1,

Ilolllthe pIlh y;t .. ll; (s ) . "iTsmn tb. space C )( C,jolnin. tlle:pOints (l". 6.0>. re. q l. ofIIle fllllcti{)n R(t) ... R(:c ...) . let 'VUein a Ric:mann surfaccified by lhc tranlfomu.Lion z .. ...' and maka it posll!b\e toUlIlsfOl' m lnlegal (~) b,-lhc chlUtsc o f varialstn:z: .. ...' into the lntqral alon. ihe ...."'.;Yinthe ...pianc' C:

IRlt .v'i"l

"CQvmnaof the lJIhue 51 wilh t wo btaDdI poinu 1: .. ' I ' e 't>The &raPh O! of the algebraic ~lWkln

..,z - a,p. - ' IX: - ,~ .. 0 (6')

the ( -sphere intO the ?sphere red_,. into ' lThe change o f variableJ

O. and'2 inlO 71 '" .. .

v - 10' (l _ 1) . r "" J:' -'l""ov, - 'v 1 - '2

lransro, ms, as can be n~l)' seen, alacbraic tqua tion (6 ' ) lero .he atsebrail:(quallon,,2 _ 7 = O. 1M COfTUPOIId1n& fIlIIPpinl . : C x C - C x C, {to w,- (T, ,,).reduces 0 2into n I .Dd is homeolltOrpbiam. and the homromorphi$m n I onlO51ispVUI by pro jtiou (2): " = I' (t}, .here t ~ ('I', . ).

Thus. we ha..., 1M commutat ive dlq:ram

")

If t ile In~oI,

IRIi:......ot:- rilG - rJd: . jR, w)c4- .

", ,;w en OQ n.. lhea bonmnuJ mappiap of d/aaram P ) mIIbk us 10 tn.n d ona IlIIl'1tOtile ill-.- .

j A(. I...)dP."

on lIIe ...sptu:n: Sl, wtme A io nuand f\lAlCUOOlo. Th is _ lJ for the ra:.loaaJlUIlOllof e eI,uq.and by lhe fo""" EllICf I:IIbl.Iitutioll

.J:-. _ fr_ :........:..J. .1 - 'JWe will come to In essentially ne'" fcsulllf ....e cOnJidcr a polynomialp(t} oflhe

thIrd dcu. ThUll, corwd an II1l

Ch. I. F,m Notions ofTopokllY"

point ,,/ te5UIIJ In a ,=han8~ of a branch of the funct.lon w, while lhlll o f any twopoints preserves l h~ bnndt, and that of &lIthe three poin ls, just like the tircurn naviaWon of che point 00, alter s the bnnch. To ' ban ' these tran sformalions, Itsuff""elIco m&lr.e the ~uu " t"1 and "J- on the (sphere. Then tad! bnndl of thefunaion w III oD~.valued on such a Jtled with th~ I;:\lll . For cae bnnd l to belransfomted into the other in the: lUtuiml way, we,rue replicas (aDd U alolll cheI;:\lU "(: and "JOb , reJJ)Caivd,., th~ cdaeJ beml slued as before . The lopolorkal5pllcen j obtainedD evidenlly the~n Jurf.a: of funaion (I). An _ tilll diffe=>Ce botwem the su rf ace OJ and the surf.a: ni iI lhac nj is topoklaic:a1lyequivakn t 10a ~ere wilh a 1undle (FII . 25, whac the cuu are fim upanded iDlO'bolu' rrom whw;h lube$ au pulled and &I\led cd&ewlse toeflher in lbe requiredway ). The nalural mapplal OJ - C is a two-sbceted aim'll ma p of SI wicb thebnndl poillU "p "1' " J ' -'r.",,,,,,,,,,,,",,,,,,:;-,,,,.-::-;:;

For t he fUl'ldion. w - ".of;. - "J(i - "J(i - ,,~ - ".), when "I, "J> "J'''~are pairwiH dirterem , we haVl: the Rimwu:l surf.a: n . whktI is hoaleomorphil;: ton j . This fDl\owJ from the one-valued brarll:bes bein, separated by the IWO t'U1S".'1iIIKI ,'I".....d the poInc ". al:liq as the polIN. of the previous cumpk (the latternot brill, a branch point) .

Hoce tllal lorqrar:lnl ratioPJ. ruroetiom OIl the Iw fKn n"

n. teodl; to elliptic buqralIh.....,. .

"Inlrodu., lon lOT""""'"

It IS "0( complieaaed ~Iher 10 Invest;,"l' the casc of an ll l~b~ic funtlionw _ -IQoti ~ l) ' " (~- '~ l. (9)

whue ' llirCpairwi5e dirrcrt~ . Here ,lI ll C\IU arc made . I.e., 'I'z.. . " . _ ...... If" IS~. and '" + l )I2 cuU l'.r1 . . . , ' . _ ~" _ .. ' ..... If,, "odd. Havinl lallmIWO rcp1iea5 of lhc z-spbc WII.h sul;h a>u , we 111.10 them alone ,he cClfnll)Ol)di llScu u : The constru ct ions arc similar 10 IboJc inIDealed in r IA 2S ..o will produce a

_ (" ) ,, -1 11 +1 ... -1 . .sphtrc Wllh __ I _ _ . , , h.nllks. n ns IS the1 2 2 2

R,cmann surfau of fUn 2 and Is an integer, and verify tlull il is .. -sheeted and10POlopeal ly equivaknlto lbe sphere,

The inVUliption of Donalgebraic lZlal)'lic functions in the ~plane aJso ICllds 10Rieman n surfaces QI1 wbid! the attal)'1 ie fWlet ions all: OM-". lued .Enrdfe 2-. Cemsid",. tbe Iopritbmic: fun

A, . 27

"

Introduction [0 Topoloty

fia;. ) 1

We sball be able 10 ba.n passm, the m ds through a loo p if we ident ify tr.e., .Iu~tOllether) {be rope-.ends

ClI. I . Flnl NO(ion, orTapa&ou"

DEFINITION 5. A knoI: whicb Is eqUoivaknl 10 . polyconal one Issaid 10 M t tlfM. AknOI whicb is not equiYalmt 10 POIy.on M ODe Is said to M wild .

EXMoIf'UiS. Th~ trivia!. tr~foil. and flaur~f.I' kDoIl ace tame. All exam ,*o f . wild knoI hai_ in rll . 31. 'rxe Aumberof loopsm thillnMllica_in.dd l.llild y Yt'hetus their size d ed'UIeS inddin.ildy while appI"OKhiIlJ: 1M poI l11 p . IIit inlcrCSlinl tbat if IlK number of kloprr; wee "I/Iit~. then lh~ knot _ Id Meqllivalml to th~ trivial one.

KlIol d assif"ttation is d oseb ntaled to propa'ticl of spaces whidJ .,.~ co m'pkmmlary 10 knOll. For C:lllampl~ , if some of th~ lopolo&kal invariaDlI o f the com-pklll m ll of knOiS X l an d X l ue different then X t is not equivalenl to X 2 (and nOIi5010pic). A useful to J)Olo&!w invari lnt II th~ fundamental &rOU P of tbe knot cornplement (the knot grOUp) (Ie~ 01. III ) . Note also lhat the set o f all kn ot equivalenceelas50el (o r iso topy eqllivalmcc eIassn) m ay be end owed wil b IJl algebr aic flntetu!'e .The Idea or l uch a struct ure lill y be , ivm In the followln, manner : call Ihe r:omposl.rioIr (Product) XI X 2o! twokll ots X t X 2 thc operat.lon o rtyln, lhcm one after th~other . The o rd er in which the y ate tied isimmt~rial ; more exactly, tb e Ilnot X I . X 2is equi.....t to the b ot X2 Xl" 1be c:ompoition of the t not equivalen ce c!.Uses1IO ckrtllCdiscommutalivc and usoc:iat iv~. Th~ cquivalmc:e c:bss of th~ trivi al knotSCI'\'C$ u tM Idm lity ckmcu l . However , all . UIIpt to solve Ib~ equationX X . I (i.~., to untie X by tyin, Ihe b OI Xl will flil exeept when X _ I .T),crcfore , t.hcknot equivalcnc:e da$se$ only form . xrn!&ro\lP (Md do DOl. fon. .IfOUP)

FURTHER Jt.EADINGThe 1'"_ lopolosbl 0ICIIiau canbe Ic:amcd ft c.. e>&II)' 1ICNfCCS. fit,~.-ic iJlt~

I.. tl;> I .......bel- of buic~ (iaoc:Iud.lnI 1-...;....,..;00 1 MId dl~1WlIlmlJlirolds) il carried: 0lIl. by Ef'ml~ in~ f1/ DtrJW1t/aT7 AIlzt"-wtia, V. 5

~'7. "~tifTopoIOu f2f1(pP. 476-'~ VlnlaI~ 1(7) by Boll)'ln'sl111ld Etranoridlmay be Quite_rlJl rt>rI h~ bq;iDncr. lt upl""l the ide:u. basic notloflSad I'"LI'lIlnas of IOPOloIY in I popuIlr mannl:\" . Abo rnl eo.w.plst>!ThpoJu lUI by OlinnWIdSlem.od Ib outd be noted .

1'bt: problems or ,luiq IOJfthcr lwo-dimcmional ,wraon 1ft' al s.o coveml ln l>O~rboIlkJ: Whlfl /$MQlhnntlt/a1 [131(Ctl . V) by Co urant and Rob blJlJ. AlfSdNlueM CAomrtrle ,[19J (0). VI) by Hilbert and CohnVossen. New Mtll~t1J/ DI"""ioIa f rom Sdtt,,1(fIcA/Iffrl('a" 1"'1, and ~ Unexpect ed 1I_,In, ilItd Other MtIIJre",.. ,kaI~ (HI. etc .llcy cacruplalN bow M~ld .slrips Ir e llScd for l'indin 'Nhm~ lnt~tilt EllIer dwaetoislic. weabo ldcd _ tedu'liclues rro-! A" A~Ioft lte &.sic Itletu ofTf:l9OIoD by BtlhYIlIlkJ' IIld Efr cll\O\icll (I6I, ..... _ from Cm.dCr (141.

The cl.aail'"atiCICI of l-.dimenslooal surracesl:s _ ed 1'ftY lill::lroq/lly in A~Topo./tIu : AMl " trotIuU_ Ij 21 (0.. I and ClI. 2) by M.wcy 1Ad.lso" !he~_

T~ [711 (ClI. IQ by seirert ..,d Tbldf.P.MCIril: WlICa IIIId thdr _plftas an: dQk with ill 1,,/I'Odw6tM I t:> SeI 71fer;Iq aMI

a-rfIIT~PI (Ct. 4) by A1allftdtOl' UIdill lbc ICIfl~at rw.etionaI-l1IU. (461(Q. m. 1491 (CII . nlTd1l'I.

An e1M1m!At)' approach 101M idea or. copoIop:,al '~fIlIY be found Ia !he 1_ boob_lolled II thc bqlrICIlaa; or tbiI wnoey. Note abo ia tllD _nealon l.m~'1Inll' Hiltorical NoI.e' 10 A.. I rfOllll !be boot by Bourbati Topotjlfw~ 11' 1 wllidI __ flI in Scc: . 1.

"IlIl:rod lltllon to Topoloa y

The "", o:oocepu of 00lft(lk:I wariablc f....etloo theory Ih.1 we fc furilld10 in Sec. . .... ybe fOtDld . tor exUAplc., .. Inland dootftenl or \he notiont ioIrod lloant)' ..,dMat~(201wbidl_~IiPcd byVOflllllCdl Slat l./IticmIJ Mol Qk'h e:aeuiDJ

General Tapol09YAI we have mentkm! above. the notio n o f melne space is insurneient fOT the de velopment of a number o f Impon anl mathematitalpl"oblems . In the twentieth call\U"Y more ameraJ CODCept o fspace bas IlI'iKn and developm in matbemati, the co nCCJ)C o fto po Josica"1 space. By now, this notion hu ba:omc uni venally ac;-Pted ainu the ' atnleture' o f a t opolosfcal space , a co ncept quilebroad and pcofound , lUUa1ly preeedes the introduction of o ther..tlll et Ure$. The lanBllQe of topoloskal soacc theory has becomeI!I cr ally accepted In aU theb~ DC madlcmalK:s which arerelated to me noc>orto f space. ThU c:hapter is devoted to the theoryof topoloaical spaces and thei r continuous mappil\&l.

i. TOPOLOGiC AL SPACESAND CONTINUOUS MA PPINGS

I. The Definition of a Topological Space. Let the re be a collection o f~ubiet.l ~ .. lUI in a se t X of an arblt ...!)' natu re so llla t it poueUe$ th e followingpropcrtl es:

(i) 0. Xe 1';(Il) the union o f a ny co llection o f set s from' belongs to r ;(iii) the lntersectjon o f any finite number o f set! from ' belongs to r .Such . collectio n of su~ets r is called a lopolot>' In X ; the set X is "ca lled a

1O/1OIOlkol Sp

"tn ll od ucll on 10 TopqlOSY

OEFINITlO:.l 2. A coU~lon B = III} o f open s.eu Is eaIled" /xJse{or i2 topoloU 1'.if for any opm set U and for any point E U . the~wsu " et Ve B such th l u (i VudV CU.TbtRfore . an y noa-mtp(Yo pen set in X QII be repruented lit t he unio n o f open5ClI f rom the base . Th is propcny dlaJactaiza baK. In particular , X equals thelUlion o f al l lM $eU (TOttI V (UlY co lkttion o f 5ClI "'; !h sudl .. property iJ, ealIcd "

I,,",II~ o f ttlc space) . ConYCnl'ly, if a see X is n pra cnted a, the Ul1IonX = IJ V.the n undu what eonditions can a topok)sy on X be c;olUlrueted sc tlw the fllmilyB IV. Jis " baM: fOC" the topology? n1Oll.... I (A C1lIlEUG.'10F" A IlASE). L " X .. lJ Y" . A co!lUin6 B .. lV.l inbaR 101" i2 ~,"f" topo/OV 1/tmd only if for " " y .' fifty v,/1'OifI B "lId,."yE V.. n ",. I~~ exisu "'..,E 8 Sitclr , /rllt X lii V., C y . n V...PROOf'. If 8 .. IV.)is " base (OT" Iopolou" th en Y" n v, is an open set. M d , by1M dd initlon o r . base, for any x v. n ...,> lMn aisls Y .., : XE Y.. C Y.. n v, .CoDYUSdy , If B .. Wool satisncs the cCNldition of the Ihcol'ftD th en the sellU :: U $I.. (aU possi ble unio ns) and the em pty SCl 0 form , at can easily be vamcd.II topalOJ)' on X (Q r 'IItilich B.= IV"I is " ba$r.

Note lbat we have abo indicaled in tbe proo( . wrJ o feonSlroc:dn l ;l to pokl l)' Ifa family B satb fyill, the f;O nd;tion of lhe theorem is l iven .

Bllt can a topo lOI Yon the $1:1 X be co n5trueted for an u bilrat)' co vering (S"J'Th e follo...m, Iheorem answers this Quest ion.TH EOflEM 1. A (1)vuf"g !S"l nQlurally tCne'ltl~J QtQJ)Ology on X , vh . the came-tlolt of n tr (v _ " 5..1, 1~/r~ r'I! K lJ on tub/In" )' finite $IIbstl from tal, lJ 0 lu:st

...for lhe topology .PaQOv. Verify Ihal the co.lltttion (1'1 $8\is fiea the cri terio n of a base . In fact, pUIV,. '" v. n V"for v.. " V".ObviDUdy. V,.'" IVl,and , lhertfore, the aiterio nofabaseis flllrdled.

Th us. the co"mn. [S.l o f th e set X dacrmilKll 11 IOpoloU OD X whow ope nseu '" aJJlhe posslbk union s U ( " S.) an d Ihe cmPl )' set .

...

DEFINITION l . The fami ly IS..J iii c:aIkd a s..bba5o> fOf the toPOlogy which kIcntniCS.

""""'=S.ler. X . R I. Set.5of tbt formS.. '2 ~:x < crJ, cr lJR l , and S, = Ir:i > ,/JJ,,/Jfl R I , form a w bbLW ror th e lopolou of th e Dum bcF line R I.6. La X _ R be an II-dimensional vector space. A base is a ccl lealon o f seuB ,. {Y ~ In R". 'Il'hcn: v... .. - Ir E R It : . / < f , < b/, I _ I . . . IIJ. f , is mej -(h coonliftate:of the:vectot..t - (t l , h . . . . ,(" );,, - (01" . o.J and b ,. (b , b,, ) an vbitn.ry "ecton III R " , 0 / < b,.

Sets lite: v.... arc c:aJItd optll ptl1'af~"p/pMJ In R -. xvrirr 2". PrO'le:rn.t the set of para llelepipedJ: described in ~ample 6 fornu bur rOf' Ihe topolo lY on R" .

"11 ls natural . fo r a topological spa, 10 51:1\ base with lhe IcaM possibk

numb of clementi . For examp le, sets II' .. ('t . tvin R I, whae '1 ' /2arc ration al.form base c:onsistina of I w unabk set of clements.

Similarly, mere is. countable base for R- cons u lina of puallelepi~..-ith ra -tional vcrtica or the fonn

...,1'1 - (x : ,~ < t , < ,-11' I .. 1, "1.wtlere ' 1. ' 1 arc n J:iona! WCI.OlS in R - ,1. Neighbourhoods. Let (X , 1) be I lopokt&ica! spKIt,andz e X an arbiuvypoiat .DEflt,m0fll4 . A Mi4libo",hood of a point X6 X ii an ,. subset D()rl C X sa ili fyin,!JIeCOlld ilM)J\$: (i) x 6 D(l'), (ii) th ere U Ul. U 1i".lIC:h dIa. z 6 U C Q()r) .

We IMY tolUider the colkc:tion of aU neiahboumoods of :;II given point thaipoucucs lhe followi8. propatiC$:

(i) the \Inion of any colleaioll of nri&hbow1loocb is. lKi&hbourhood :(d) the interwaion o f ful.itc number of Ileiahbourl\oods is a ncishbou rhood;(iii) MY set conlliniq rome neighbourhood Ott> Is nrighbourhood of the

point )c .THEOREM 3, A sub.wt A(A *" 0)01il lopoIo, ka l space(X, ..) uopeni/anr!ontyif It con faiM somt nt i, hbou,hood of tad of its points.PROOF. Let A be open , x EA . It is dcar then IhAI A is a neigh bourhood of x ,Therefor e , A con talll.l ll neighbourhood or any o f Its PQinls. Let for any x itA ,there u lSl a nei&hboumood of the poin t x , Iyln, wholly In A . By the definit ion of ane\lhbourhood , ir co ntains some open let U". x e Ux' Consi der the union U U"

...

of liudl sets for all x EA . II' is open; A C U U" linee any point of Ihe Jet A...

beloo.., 10 .U U". On Ibe other hand. we ha~: UII C A fOl'" every x , r.e w

u U" C A . There fore. A :< U U}< . and A is open . x." "0'"

NeiJhbouthoods Me wed for sepaJalin, points rrom cadi ot her.OF1l'o'lTION , . A I~&ical spac e (X .,) ISsaid 10 be HtnUdor/ / if Cor an )' two dif.feraN poInru .,. in iI. there are neithbourbooch U~)andUU') of these potnu we h111111. U(.t ) n UO') ,. 0 .

A IOPO\o&it&l space ~. ") equipped with the trivbllopoJoay is not H. usdorfCifit (OQW!lsmore Ihan 0l'Ie point (YUlf )'!).

'Ibt$e ptopenltSohhc:neia,hbour hoocbor . pain! (wbidt arc now dedared to beuiomJ) are oftee ,,2

Introduction to Topolo8Y

laim some Q,, (;r) then U Is also a neishbourhood of the point x; (iii) for any twond sh bourhoods n", (X), n.. (,x) o f the point;Jr, their interwction n.. (,x) n O"'~) isabo a ncighoo urhoJd of th~ pointx; (iv) for e\let)' neighbourhood n'(x) of the pointx, th ere is a ~lghbourhood n (x ) C O(.r) S\I

"AsJodll11na~ o pen lee U o f the spaa: X willi Its IfI!.l&e I(U) W>dcr IIIl~rp/Iism j : X - Y w abl i.l.hd; II bi jea.i1: conupon dcaCf:; bet ween thetopoloJies 011 lb.e spaces X and Y. Henu. ......y Pro~), of the~ X Kaled if!tmnS or II lopolosy o n th is $pC is also nlid for the space y ..hid!is~pbX III X, and is il EnilatJy " ated in terms o f the lopololY on Y. Th IQi. the~ 1pa.teS X and Y poue$S idc:nUca1 pro~a and I.I'C tndi!lin&viIbabk from thjs poirn of kw.

The: propnties of lOPOkIsicaI spe.ca that are preserwd undcT bomeorl:Iorph~arc ealkd l~iotJI~ia . Note, in th is COllnec:tion, lbal the main u.sk o flopolOl)''''''' for a lonl tim e (and st UI remains partiaD)' utIso1YC'd today) 10 discoveran dfeatvc method o f dm ill.luu.hing bd.""'em nonhomeomorphic: spKeS.~rcisa.' " . Sho w thl.! II homeomorphism dacnninC!S ill oom:spondcnoe bdWf\ the b asesand . 1Ibbases for homeoIIlorphie spaces.9", Show that ure homeomorphiJ:m relation is an tq uht.lcnce relancn .HI" . Show thll' the Interval ( - I, + I) of the numbcTline II homromorph i

.. 1lll1OClualon to Topolou

THEO ll.EM 6. n co f/Il1ppirll/ l r : Y - Z is ron,i"lIOlU,P l OOf. Lei. 7, be . topo!osY on the Iplce Z, and W e' T Then Vly)-I ( W)_'" f - I(If') n Y and l inter l ( W) E r we will ha ve III y)- '( W) e "y Mr(l;u .1) 0. Show that UI open set in l subspace Y of . i Pace X is nol~y opm inX . Conslckr the cases o f X _ R ' , R}. RJ ;,tte:mpt Ihe IOlDIC qucKioo rot closedxu in Y. Pro!: . priori !h lll. any cloKd ~ F" in Y is of u.e form F1' '" F n Y.where F .. . closed K1 in X.14 . Lc1 A . 8 C X be elo$ed :leiS o f . topoloP:al fPllCc X . and In X _ A U B .Thcn l mlppin. ! : X - Y is eo ntln uoWl If and only if/I ,.. : A - 1', / 1, : 8 - Y;u"c co ntin uou, .

We no.. introduce ano ther importMl notion. A INppi", I: Y - X ii c:alled anembeddi116 of Y into X if (il I is eon Linu_ . [Ii) i : Y - i (Y) is . homeomorphism ,when i{Y) C X is . SlIbspaot of X.

Embtdc\i.Dgllfc useful wMn..-e inlaid to 'sir!sk OtIl' subspK'C Y c X of the:ambient iPUC X and. to e

Ch. 2. em...1TopolOlY .,

THEOR M 1. Tlt t: IOpo/OO ~. t:O/lJ1111r:lnf is Hausdor/ f .' 100f. I..ct X __ r- TlKn P(.r . y) .... > 0 (b y I p ropt:n y o f a metric) . Settinge '" i .we.~Udcr D . (.r) . D . O') . II is t:as)' to s.how Ih a.! D.t-') n D. O') ,.. 0 Infxt. if wt: asswncd!hr: con trat)' o-e: would have

...... .. p f;t. ,. ) " p(.r, d + p(t , ,.) < - ... - - -l , l

for apoi nt (E'D. t:r) n D , {Y), which is impossiblt:. Anotht:r an d equivalmt defi nition o f open St:\$ in a metric space can be given .

DEFINITION 2. A set U '* 0 is 0{Jt: /f if for any:t e U. th ere is an open bail D. txlwith tbe centre II x which l i e.~ wholly in U .

Note Iha t we. defined a lo polo gy in R 1 in preci$ely th e Sli me manne r. IIIId,therefore , it eo incid Cli wilh tilt lopo logy ... generaled by the Euclidean memc p o nIhe plane R

'

. The vcri rlClllia n of the equ ivaknet: o f th e IWO defi nitions is It ft 10 lilt:"",U .

Consld u a mappina!: X - Y o f a met ric Q)aCt: (X, p I) iftlo ;0. mtl nc tp aC'C ( Y.PI)' Now two de finitions of the tonlillully o f lilt: mappiftl/ r:an be C1vet1. m .. as aJtt.appifll. of metric and u a mappin, of lopolosical spaces. The$t: two dc filllllonsarc. equivalenl . vU: . Ihe folloW'illllhcorem is va lid .

TlfEOREM 2. A moPPIJI,!: X - Y o/ tI _ rir: SJ]tIIC.T ./f;tO -c e whenII > N .

Conside.r an ope n ball D.VCxr}) in Y and denot e 11 by V . lIS inverse Image1- I(Vf ) is an open ser in X due 10 the continu ilYo f! . mceecverXo er ' (V. ). Thepoinl ..-ob elonp to/ -I(v. ) tOjelhe r wilh some ball D.Cxol o ( rad ius

.. lnl rodll,:tlon 10 Topolol)'

belong.s (0 Y together with a cert ain open bllll, W(: co nclud e that f(,rft ) E Y andx" 6 r '(Y) bclJinnina with some number ". which is co ntrary to the a.s.sumpt;on .Th us, the mappins f is continuous in the topologies of the spaca X and Y whictlwere induo:ed by the metri cs. _

2. SpaceRI! . We shalJ consider an im po rtant eJlamplc of IImetric space, i.e., theE",cl ldetllf s~

R" - ((E . . , E,,), - 00 I) is defined similar ly to the metriconRl :

( I)

W~J( .. (fl ' . Ell)' Y '" ('" . , 11,,) arc 110'0 .rbill'a1y vect ors from RItLet IUveri fy that this il a mie Tie . Evklently. Properties I , II , III of a met ric (sec

Sec. 2, 01. I) are fulfilled . Con sider Praputy IV. II is rcqu~d to prove the inc-q uality

for ar bitrary real numbers E,. "'_ fl'; '" I . . ,n. The proof is broken into twolemmata.

LEMMA l (THE CAUCHYBOUNIAKOWSKY INEQUAUTY). For any rtal num~n~I' ", i = I, . .. n , th~/o/lfJwinB inrqUD/ily holds

PROOF. For an arb itrary real 1\. we have E (fl + 1\11;)] ;;, 0 , whlffi~

,. ,

E E: + ~ 1: E:/l/ + >,Z E ,: ~ O. Consider the left-han d side of the in-1 - I 1 _ I 1 _ r

equall tt as I polynomial in 1\. It cannot have two different real roo ts. Therefo re , itsd.Iscril:ninant is lIOI\opolltive. Hence , the inequalit y

E: E 'If ,- ,

Ch. 2. C."",al TopolO!y 4?

LEMMA 2 (THE MINKOWSKI INF.QUA..L ITV) . For Ilrbitrory N!UI numb~n ~I' " I.I '" I . . . . n . 'h~foJlowjnll inequality is valid

( i: " ,+"" )'" ( i: 'i) '" + ( i: ,i)''',0 .. l I. .. I

PlOOF. By using the Ca \lchy-Bouniakows lr.y ineq ual ity,

r Q', + >1,)2 '" r (~ f + 2l:,l1, + ,,~); ~ I , - ,

+2 ( 'in r< r e r r ,,1 + r ,",. , ,. , ,. , ,. ,[et r + ( 'r r'i r " ',-,

an d by tak ing the sq uare root of bolh sidrtofthls inequaHIY....eobtain the requiredinequalily .

We ca n no... complete rbe verif>eatio n of P roperty IV of the met,ic. Usins theMinlr.owski Inequality, we o btain

( . )'" ( .c ,~ ,(~,_f,)l + O ~ IThus, p is a metri c o n R". Ut xo _ (~y. . .. , ~~) be lhe ce ntre o r a ball D~""ol, an d x .. (~ ;. . . ~,, ) its

arbitrary point, Then the coordina tes of a po int X sa lisfy the ino:quaillYI ~I _ ~Y12 + .. + l ~ ~ _ ~~12 < r2, (2)

A ball in R" is often denoted by U;""ol an d called an oJNn 'I_disC. A set of pointsxwhose coordinates 58tlsfy th e unsmcr inequal ity

I~ i -~~ I '+ . + t~n -~I ' ,;;; r2 (3)is called a cloud ball (d OSlid n-dl$c) TY:V r;). The (n - l)-d imensio nal sp hereS; - 1(.l'r;) wilh radius , and centre at lhe paim xl) is d efined by the equ ality

11 _ ~Y1 2 + .. . + I ~" _ f~12 = r 2, (4)We will call It th e boundQry of tht! disc 15'; or JY; .

A metric on R" can be defined in other ways , for e: "" , )' ) = rna" (I ~, - 'fII J.

,. " ." '"

llltrodlldioo loTopoloty

~ I - _Describe II ball in R" by means ofm~ (' ) . Show th . Ihc E lJdid earmel ric and metric: (S) ind uce Ibc same topology.

ConsIdft" the complC'Jl .. -d immJionll1space C" ;c- .. l:t ; c .. I' . .. c,,). c,. .. x. + iy,.. x,.. J',. I! R I . k .. 1 . . . . Il l-

llIe ma.rk on II iI. introdl.ll::l ln the same: 'II'lIy as in the real case ;p(t ' , z " ) .. ( Ici - :t:i ' 12 + . ' . + I:t:'; - :'; -11)11lI.

where.t .. k ; . . . t~ ). e ' - (t j ' . . . , c~ ') are: elemen ts o f C" . The -..mc10POlosY is detmnined by the: marie

p(t .ZH). mLll ICk - t k 1... . I.

We no w formula le: a condition for the conl inuilY o f mappinp o f Euclideanspe.ees. A mappingf; R" - R'" associa tes each poi nt (fl ' .. . fIll wilh a nainpo int (" I ' ... , 'I,,>. so that we:a n wrile

'" - f t U I" . . f ,,),(6)

...... f.(f l ' . . f,,>.wheref" " I _ . , m ia a n~ fuoction o f II variables . This fUnCiion deter-minc:s a mappinzf,..: R " _ R l by the:n de

'Ii -f,(f l' . f .). (7lIt if evidml that th e COIUinwty o f th e mllppill a /' is cqu ivalml to I~ co ntinuit y

of the nwtlCfical functio!t JjU I' .. . f.> as iI is dcruted IJl analys.is.CaU mappinl (7) the i ..fl\ component of the mapplnlf. The mappina l is det er-

min ed by spc:cif)'in l aU itH mponcnts f ,. j - I . . no.THEOREM). A nuzppilfZ f ; R " - R- is cot/' ;""ow if llnd 0 ..1)

"..

""

Fl... n

and D1 is th e inc.erior of th e lll'lil circle in R 1 (AI. 32) ,. - (~ I .h- h) E S1 .. , J1 " 1(\.h.OJ e01.

Th e Projcd>on U I_h. h ) - (f l ' h- 0) is . bomeomorphism of lh e ht;mis pha-c S~UMI dilorpbisrCl (8) is also dcrlrled o n S'" - 1, and lh alr II SO _ > - IS" _ 1- Thus , 0- - I is homeomorphic 10 St'. - '.

We now csu,blish llnOthrr impon an t~(omorph ism .THEORE..'Iot 4. T1l~ disc D'" .. II ortWomorpJrk to /II( space R"'. m ;;0 1.PII,OOI' PUllinS m .. " - I . we use th( pr( vIou s constructlon. We tl'3nslace the. pace R~- I, II "" 1, so Ihal ch( orls in of coordinate:s 10

"IA(foducl!oa to Topolop

ia& ltl iJ interstttioo poUK 10 l ilt: point Jf, _ obt_ the ltIappiq +: ~-I- R" - I , h 'CD by tbe l'\Ik

(f l' ... .f.l - (!l ......!L:..l . I) .( .. (.

TIW IlUoPJlin!. Q il ill euy"'o .airy.is.~. The M1papos;cion ottbeho_lIlOfPbirms

.r 1 : D" - I _ R" - ' ,If > 2.)'idds the rc(luired homeonloc'phhm.. ~iws.2 . Sllte . crilUion of the l:OlItinully of. mapp il1l / : CO - C'" of oomplell' spoeet'l .J . Prove lha t C" ill homeomorphic to R 2II,4"'. Prove that the balls in the .pace R '" which are defi ned usins melrics (1) and (5)ar

"Naturally, we rdel" 10 the empty set as an open set. This collectio n of c pen subselSin X IR is a IOpo]OI Y and d cnol:cd by 'R'Eurcist 10 Verify tha t-'R is a topo logy

lnlrodllCtioA10 Topo !olY

,

'r:t,- ,. , ,

""'C-31

line: /lib will be all mlO $Cpnmu wh icb COIllP~mc:nl each nthn to ''''cles and lie or:th e:oppl)lltc: sides of the red.ullJe.. If is d ue' thaefore:d Ull it i$ Il:CSSarY10~the: complementary searnenu loaeth

CIl. 2. GenO!aJ TopolOSY"

J.Mappings of Factor Spaces. LttX,X ' betwotopologi~al spaces andR,R equivalence on Ihem. Consider II mapping/: X-X' . We ..... ill u,y lh al thema.pping / prtservrs ~qllill(lltn If it roue.....s from x ~ r thal/

"la ll'D

ct>. 2. Gw cralTopoioSY"

THEOREM 4. The $pIJ~ jjtt I S" - , IShomeomorphic 10 Ihe 5phu~ 5" .PROOF. tn Item 3, Sec. 2 , it was shown that the disc D" is homeomorphic to meclosed hemisphe re 5"... This homeomorphism is the identit y hom ernnorphism onthe eommon bou ndaryJ.S" - I) of these SCIS. Therefore, the equivalence relationfrom D" is induced on S~ , and D" IS" - 1 is homeomorphic to~.. I S" - I by th"last theorem.

We shall now sho w that S".. 15" - I is homeomorphic 10 S" . The inclu&ion1".. C SIt1.5 natural . Denotethe South Pole (0. 0, ... , 0, - I) o f the sph ereS" by _.There must therefore wst a continuous sur jective rnapp!na 1'" ! ".. - S" such that..,(5" - I) _ . , and ", IS".. : SIt - S" , l- j is a homeomorphism. The larter can beeeestructed, for exampl e, es follows: if x E !~ and x '" N fI" is Ihe North Pole)then we draw a two-The produ ct of th e two homeo morp hismsDn IS" - I _ 5"..I S" - I . ~/S" - I _ 5" ,

is the requtred homeomorphism .

4. CLASS IFICATION Of SURFACES

1. Surfaces and Their Triangulation . Let uSreturn to our invwigationof ciosed surf ac:es. The defin ition s of a topological spa, f;;u:tor space, homeomo r-phism of topological spaces givco abo ve and the examples eonsidered make up a1I01id buis for the proof of the theorem mentlonet1 in Sec . 3, Ch . I. It states thatIJIY closed' surface is topologically f:9.ulvalent to ilTI MJ!- or N q - surface : i.e. , to asphere with p handles o r q MObius strips &! l,Ied t6 II. Here, the correspondine no,lioJu will be roade mor e precise, and the proof of the ..hove_ment ioned theoremgiVI:O .

A. topological space X each of whose: points has a neighbou rhood homeomor_pl>ic to the open two-dimension al disc wlll be called a two-dimt nslonal manifold. Itis more convenient to Mudy thC5e spaces ir they are broken into elementary pieceswbieb are topo logically equivalent to trlan&!es in th e two-d Imensional Euclideanplane. Let us make thls representation more precise .OEFINITION L Call the pair (T , ...), where T ;s a subspace in X , and.,. , .& - T is ahomeomorph1.5m o f some tnanel" 4 C R10n T , a topologkD l 'rllJngl" in X .

If the homeomorph um \

"hllfod llClioa 10 Topoloay

... ..

lopolo&ieal t rlanaJe T. For unlfonnil)', II is also con~nit'1I1 to can lhe $IdC'J of theInanale'" the celan.

We now define an o ric:lltatilln o f a trlanlle. Different or dered tripla of point!can be formed fro m tile ven~ o f d. We coruidct IWO tr iples 10 beequivaJenl if onecan be oblained from the alber by . cyd k permutation . C\c....ty, t~e arc ~aetlytWOequivaiClOll classes. A trltln,k '" if ori#Ittr! i f one o f th_ equi vaie oce elasmis ra ul. A l opdo, lclll ,,,"r/w(T . ,,) is said 10 be orlmttd If the triantk '" isoriented. It is ob rious lbat orientilll a t rian.&!c 4 is eq ulvalmt to siina: ""aiDdirection to the c:imlrnnaYiJltion of its vtttka (cklcln.Uc or coWl tadoc:lr.wisc).Th is Qr CllmAilviDtJon di rect ion drtc:nnir\e$. via the bOlM()lN>rp/'lism .... c1ittioDof lhe circu mnaYi&alion o f lhe venicu o f the IOpologiClll triu&le . l.e . an orienla -Uo!l induced by lbe homeom otJl/ti sm " . An o(i entat ion of triAnaJe obviouslydetennlnes the orientations of ils cdtl" Il.e., order ed pain of Its ven ieaj .

Note for Ihe futu re tha t an ori ml81ion of lUI lI-gQn ud lUi edges when " > 3 isdel'ined in pted$ely the same way (by aMna an orientat ion for tbe eircumnaviat ionof Ihe vert iceS).OEFlNmON 1. A rmile SCI X _ l(T, "'i)-"' . I of topoloakal l1i~cs in X Ihal

fulm the eondiliOll$ (I) X . U F j and (2) lite intef~ of all)' pair of

...

IrianaiCSfrom X is either cmp! 1 Of cciDcidcs with thei r rontmon vertex 01 tommOllecIa:e . is called a IrlDnp/tltiQn of the l'IO-dimtftsionaJ man ifo ld X .

A manifold for which mere exUl$ a trianlu latlon is ~id 10 be frian ru/flbft . Ifany IWQ vmitts o f the tr lan,lcJ fro m X tan be joi ned b)' a PlI!h made up of theeda:a. then we o;aU X oonntc:ftd.

In rIB. 4(}, VI uampk of a trianlulallon of Ihe .sphere 52 co nsis tina of eigh ttrianaJes is shown.OE.FINrT tON 3. We c:al.I a connled. t riClau lable. Iwo-dimenMorul mamfold a d os-N swr/ llU .

Note that th e u.amplo:s o f e:1osed l unacies wbldl ca n be triansllb":d HlIOtollOlolPc:al poIy&Oftl aDd wh klt we COJUid~~ in Sec . J , 01. I. arc: examples o fdo$cd lur faCCl in the:ICIIX of Definition J (to pt'Ove th is , it 5ufl'ic:c:li 10 l riansu lateIhe polyCWU).E:MrriH l - . Const ruci t rian s uhtt ions o f a to rus and a project ive plane . Verify Ihac

'~ey are:d osed sur fa l:eS.The lopolopcal prolXnies o f . clo sed sur face are dele:rmined by Ille stru e: lure o f

its lrim aulation. To ;n",""ls ale: lite: Iria ngullllion, it Is eo nvenlenl 10 con sider ilSIdItmatk represenl ation on the plane . Moreove r, plan e tria""es l1/and lite:' nveuelrDqes or the tran&!eS TI e K ..,.y tK: coll$idered 10 belOfil to the sameplarte and bellltMuaUy a dusi\ 'C.

We will dacrib:e s..cb rqoraetu ation . Let (TI, " ,J, (Tr "') be IWO Irian&IesftOtllK, and T, n ~ _ tJ the;'" com mon cd&e; lettl; :a " ,- l (4') and tlj '"' "i I (lI ) betbc ~DdirlS edScs in 11,. 4/" The s1wna bomeoft1Ol'llhiYn is d etltlcd as

"11 - "j l;I../ I~; Q/ _ Q;Thus. the 1riansuill ion K can be as:so

60 IOlrOOul;(;O/lI OTopoloay

,

'D'----------, ,-t 0 t~~

I ,, ,- ,

.... - --71' - --- ;Fit!:.

Note th ill if the the location of a polytlon Q, on lilt: plane is altered by Ihomeomorphism ..,. then we act new homeomoill hisms (OJ'''u-at IJ tha t s:Jue iuedges, and which weshall not distinl Uish hereafter from the hOmeomorp hisms (YOu! .In particular , thc family

"J. The Classification o f Developments.

DEl'INI~ , . T wo devd opmcnts Q and Q" aft A id 10~tqu;vlllr"t if theW f actorJ!!"KIIlS~ bomeocnorphW;: .

We_lnlTod~ lOme elemental'J'~ions OVCf dcdopmel>t wni(h willll1UlS!0I11l it into lUl cquivakm (mIO.

SU8OtVISlON. l..d. then be an II~ 2 1 f" > 3) In 'lOns Qi ~ Qi ' :and moYt the polYloruQi -.ui Qi '~ oOllSnuctinl a new development Q from Q by rqJ 1adna 1M.pa&Uon Q,b)' IWO pol Y&.O

!l>lroduellon to Topolosr

ere. See CII. I fo r Ole words of Ihe"~_es 9 f ~ltepmllion ilIId triana:1e wblch are thedevelo pments o f. hM dlc and M6bius st ri p, rupectivdy.

Ha >inS deooI ed the ed.&es of aD Ihe pol),lOm Q, ....e obtain a sa o f wonls(wCQJ\. ....here ...CQ,) is a word dc nodna Iftc 'Vui.llS ruIc: Of lhe polygoo Q" In ldelidon. tllle lcUef1l in Ibe won!. ...CQ,) arc ....ritl Cft In the onier in whicb ""' tM ax-respondinl sides o f!be poInoo Q,.eeordlnl lo itl o rien tal ion . It i5d ear that the Ia-

~ted sa of $YRlbolic 'IOOrds !o-CQ.)I determ....es the "'elopmcnt Q .Two IIWn types of devc:lopmcn u can be t.in&kd ou t .

OEFlNlTIO!'l 6. A T'ypr I auton icrJJ~r is. dc'Iocloprnent consisti.Dt of ODepoInon detntnincd by. word who fonn is . - I or

lI,b"" Ib, IlIP }!'i 'b i I ... lI.,b"p;'Ib;' I. ", > o.OEPINrTlON 7. A~ItCilnoJt/all dewIop_1 is ~dopmenl~ o f llIlrpot)'lon with word of the: form " I""'}!'! . . . ""pM' m > o.

We now formulate the basi c resul t.

THEOREM I. A II)' dtwlopmMr Is ~{Wl/enr 10 "~ / ()T JJCtmr",icDI de...m,p.mornlocclHdin, 10 ir" orienr"bility lH non~"'"bi/ily.PIIOOP Two mt'IIIrks II nut. To bc&in with. it Is eaJ)' to see tbat by s1uilll, litedevelopment C(ItTapondios to a triangulation K o f a w rfaec X ean be redllocd to ,development a:msisl ml o f one pol)'gon . We shaD ther efor e consi der onl )' lb i.l kindof deve lopment. Sc

0..2. Oc:ncra l T opalo l Y"

1'"'8. 43

Furthermore, if slillsome vert ices which au not equ ivalen t to A remain, then we~t lilt whok= p rocess descri bed until we obta in a dev elopmen t with th e requiredpropmy.

Thus, we shall assume fro m now on th at aU the vertices In ou r development arcequivalent, an d thaI th ere a re no combinations of the fo rm QQ - l' ln h.

(2) We m ow now th at two similar leUen in the word of a d evelopment can bellwaysplaced IOSC1h~ . In r...." lei two let ters Q and Q be nOI placed toget her. Th endraw the diaaonal. d join ing lbl' initial po ints of th e two edges

.. In"oductlon 10 T opology

o

co nt rar y to the equivalence of all yenil:e$ of lhe development, since 51,1cl'la situationis pos.sible only j f the vcrtlce$ A, B of the edge (l are not equi valent (Fig. 46).

(4) Thus, there au two pairs in our word , Q, /1 - t and b, b -',

t hai &epanue ea.:hother . We now demo nstrate that Ihac: fout can ee always replaced bycombinalionlof !he form -'J'X- 1y-l whilst kecping1hc conditions of u eps (I ) and (2). First, jointhe origins of the edges4 ande " with th e diasonalx and make a subd ivision a100Sit ; then glue a10nlllhc edge b (Fia. 47). Join th e ends of the edses x andx- I in rlK:polygon obtained by Ihe d1aaonaiy, subdiVide along)' again and then glue alongl(Fig. 43).

We oblain a developmen t in wncse word then is the comb inat ion X)'X - 1y- 1 ig.stead o f the letters a. b , /1- I, b - ' . If combin&lio ns of the form ee - ' appear aflel'these operatioM. Ihm. th ey are removed by con volution while combinations of therormdd and a/c- 'd- l an: not separated. 11111.', the sit uat ion reached after steps (l)and (1) is pr~rved.

By applying th e cOnstnK:\i01l5of steps (1)-(4), we ha~e'tnlIIsformed the ori8iollword to that oonsist ina of combinations of the form }C)lIe- ly-l and aa . If there artDO aat ype combinations in the word, then th is is a Type I canonieal development.

(5) If th en: are blllh 1e)lJt-ly - ' and ao type combinatio~ Ih en th e wold can bereduced IQ th e Type \I canonical form ill the following waY. Join th e commonvertCll of the edges u and a to the wmman vertex of the edge s )l1lIIcIIe- I with Idiagonal d. Subdivide along d and glue along (l (Fis . 49). The two palu of the

Ch.2 (>

..

In lh~ proof of the theoftm,~ba~ soown that if P handks and II "" 1 Mobiusst rips arc liltedtc Ihe sphere. Ihe!lw obtained surface Is non-orienlllble and o ri lleN" .. . 1)'pC.

1llt devdop:ncnl dassificatioo lbeDrcm Iuds us 10 the conclusion that Anyc\o$ed. surface is homcomofllbic: to. main sw ratz: of the type Mp N." To makethe reNh mon predK, COIlSidn' the EuJtt charact.erUtic of OW" sur face. Let thed eeompositioll of _race x contain CIlo vertices. 0, cdaa and 01 imqa ofpoIYJOr1l. 'I'M number x (X) .. 00 - al +

Ch 1. G. ... ral Topology"

The ileCond pa n o ( th. theorem Wall explained in Sec. J, 01. I (It em 4) an dabove. This explanation could ha ve been consi dered to be the proof. had thetopological invariance of the Eul~ characunst lc X(X) (oc- an arbitrary closed sur-nee X (we have o nly don. 50 fo r X = S2 ). an d th e faetlhat Mp an d N q are nothomeomorphic . when q = 21' .p '> 0 , been prov ed . Th ese fads will be estab lishedIn Sec. 4, 01. Il l, usin g th e id ea of th e (undam.ntal s roup of a space .Exerci#.! .3. On.... the diagram (or sl ulns a su rface whose eallon il:al. developm. nt has theword

Qlbtoj 'bl lazP'J!tZlb i"topflJ IbJ I.4. Oaw the diawam fOT BJuin s a su rface chm cter;zed by the word Q1Q IQ-PPPJ'Imlicate the type and genu s o f this su rface,S. Veri fy that the following closed surfaces have the Indicated type an d genus:(l) the spher e has Mo - No:(2) the loru s (t.e., the sph.re with on. handle) M I ;(3) the double to rus (i . . th . sphere with two handles) M 2:(4) the projective plane N I;(5) the Klein batik N 2.Draw the diagr ams o f their dClCOmposition s.6. Call a topolosical space in which each point has a neighbourhood hc meomcr-phlc to an open inte rval of the number line, a ont!-dim~ruionQI mtmi/oid M 1, Callthe dcccrnpoiit ion of M l in to al'Q which arc th e topeloSica l imq . s of the line .IC8"'ClIt 10. 1) an d who se ends are adjant to each other (i .e., med at vertices) af,*,nglllQlio n o f M 1; we UIIume th at M I consists of a finite number of arcs.Prove tha t a trian lulable manifold M I is ho meo morphic to the cir cumfo:ren ce S ' orsome o f its replicall.

S. ORBIT SP ACES . PROJECl'IVE AND LENS SPACES

I. The Definition of an Orbit Space. We cort.lider here imponanl ex-amples o f f~()f spaces arising when groups act o n topolQgic ai spaces,

Let H (X) be the set of all ho meomorphisms of IIto pologieal spaceX onto itself.The p roduct o f two ho meomorphisms h I and hz is defi ned as fo llows:

(/I lh~) - h.p. I(x) In add itio n. for each h E H (X) , there is an inverse mappingh - t e H(X) and hh - I "= h-I" = I x. Thus,H (X) is a mu ltiplicative grou p (non -comm uta tive . gen erally Speal:;inll) with the iden tity element I x .DEFINITION I. We will say that an a bstract S!"oup G acts (Jro m lhe Ie/t )on a sJX1Cf!X if II homeomorphism o f the arotI ll G into StOUp H {X} is given ,If G acts on X then to .ach , e G, the re correspo nds h, e I f(X):

t - h, ; 8 182 - h' lh' a,g -l - (h, )- I.lo - iX'Let x e X be an arbit rary poin t; the ser U h,(x) is ca ned its OI'bll and deno ted

by OJ(' a_a

.. lnuoduclioft 10 T opol lllY

~rr;1SI ' - . Sbow that two orllilll 0,. 0 , either eoiaadc or arc disjoint ,TIle: lui SlatCiDQIl enables liS 10 intn>dlltt on X Ul eq"jYakncc R , x ! ? ..

.. 0.. _ 0,. i.e., ~Jl.nd , bdon& to 1!Ie same orbll.DEFlNmON 1. 1bt rlt\Or~ X IR is caIkd the orllit spo of 1M /lrtJUP G .006cnoted by X /G .

This method of C

,)

0>. 2. Gcner.ll TapolallY

iKlSon it according to the ru le ~ ...x .. (~Ia~ I' ~"h.....~~~ .. I ). ThUs,group aWI be identified Wilh th e un it circumference S l in th e complex plane C . He nce. S 'ICUon the coordinate {, e C, and the o rbit o f the poinl tj ln C is the clecumferenceof radius 1 ~ 1 1 if I~I I "" O. Therefore. Ihe o rbit Ox :::> (r'"xj (0 " (l>

"6. OPERATIONS OVER SETS IN A TO POLOOICAL SP ACE

In this $leClioo, we qain tum out . tteDllo n 10 the in_iplion o r tlle pro pettieso r topolofPcal 'PKa and conlid er the o::lcmlu and int erior operations and the boun-dary operators on a K'I and also two l'IOlio n. elol.Cly re..ted to them. th at i,. the cce-c:ep ts of lim it a.nd bound lU)' points. AU thcse Ideas gel'lcral il.C well_known conceptsof malhematkal an alysis .t . The Closure of a Set. Let lX, Y) be . tOpo!o&iu] lpace ,DEFlNITlON I. We dd ine the e!o lun:Ao f . setA C X to be the intcrsec1ion of alldoso:d $l:U coar.a!nl.n& A .

1be foDowin.5IlJecncnU ate obriow:( I) 1bc~ Al$ the $tIWIe$t dosed let eontainins A .(2) If A closed th al .i A .A dosed sct CllD be charllClmud by lim it poinu. whicb _ do bdow.

Dl?FlNtTION 2. A polntx E X Is sa id to be limIt one:of. aivcn set A C X If there11111 least one po intx ' e A ot her IhllllJt in each neighbourhood D~) o f the point x.&e~iK I " . Verily thll! th is deflllilkm can be restri cted to only ope n nc~bourhOO

"Then fo r any DeiahboUrhood v(r ' ) o f the poiDt)t ' such lbal 1'(1") C O(r). wekne V(r ' )nA _ 0 . thc:rerlX". x is notalirniIJlOUlIOrA ~O(lr) n A '"' 0.1'bIu. OU') C X '.CA U A " ) . and bec:aw.er is vbitrary, the 5C:t X '(A U ", ' ) is.,....Eumx 2". (I) Vcri(y WI (A UB)' _ A ' U B ' , (A nSf CA ' nB' and(A 'S)' :::>A " '8 ' ,(1) Lei X .. la. bl be .. spa cc o f two clement' cquippro wilh lhe topology eons;stinsQf lllc thte" sets: 0. X , la) . Gi..." an ex.1lmple of a KtA C X fOI wh ich the ind usia n(A T CA ' is 001 valid .

We shal l now prove .. buic statement abou l lhe structure o f the:: closure of .. set .TIlEOR.M 3. X - A U A ' l or anyMt A C X .PJOOf'. 8 y Theorem 2, ee KI A U A ' is~. Therefore. by the d efin ition of.. do$urc, A C A U A ' . On t he other hand. it is obviou.sthu an y clcna1 Jet con-tairlinl A also COIltaUu all limit poi nu o f .4 , and ma do!c oonlairu A ' . Heoce ,A U J!' C A. ThIlS, X '" A VA ' ,EnrfiM 3 , Let A W 11v XI oj""tioll4lpoi1!u on 1M. ,"1$trai$"t liM R I.~l.!lacii - R I

If. topolopeal lPKC X h., .. eounable subset A ..mote do$ure urinddcs withX, Ihen It Is~ to be sqxupble. (I is cu:r to verify that~mty is a lopoloPea!P!VP"r1y.Extrf:ius,4" , Sho w that the Spate R ~ . the dise Dft an d the sphere S" - I are separable .j " , Verify the followina properties o f the closu re o~fation :m '" A U B,A " A,~C A n B,A ' Be A'::B,6", Let Y be a $Ub~ of a topoloJical space X and A sulurt of Y. Denote thedoIure of the set A in the SIIbspaa: Y b:rA y . and Ihe c10lIIfCofA in X by A . Showthat Al" - A ny.DEFlNlTION 3. A polnt x ~A is oaid 10 be isolDlftI if there is . nci&bbowhood 0(%')of the point x JUdI that it docs not contain lilly points of the Jet A olha than x ,

A poin t X 6 A is iloJated if and o l\ly If x e A ' A ".OFt1'4ITtON 4. A set A is said to be d i$r:,c,r If each of its points is isolated .z.The Interio r o f a Set. COll1idef t WO other lmponant no tions co nnc:ctedwilh that of neighbourhood.OeFINITION ,. A poin t x e A IS call ed lUI iIl/mo, poiflf of a set A i f it hu nci,hbouthood fl (x') sUU- Co lUider A _ [0, II, Ihe line sca,mc:nl of the l u i st rap line It I. II IScur10 sec lhat (nt (0, I] - to. I) .

TIle operation Int is dual of the dosu~ operation. which follows flll m ilS ","0-pcnics as alunciated In Ihe fol lowin, thcoran.

"lnuodu o' ion Co T..."oloV

lliEOREM 4 . For 1m }' XI A ex. It't' /IQ ,'li': (l ) Inl A ~ (IJl ~/I .J1!r; Ol lnt A is IIwJa,.,ur 0Ptll st.1 COnlD;nM ill A ; (l) (A IS O/Hn ) (In cA .. A ); (4)I,l"G InIA) (I' E A orfd}l is 110 /" J"n" polit I of X 'A) : (S)X' ''' = X , In tA .PIl00f. Properties O){l) au aImasl e'ricknl . We wit Y'tTif)', fOfcump~. Propm)' (I).Lei x E Inl A . Then Ihne is an ope n nclshboufbood U(.l') of Ihe paint x $uch thatU (I') C A . The rcrOft:, IJlI.A is IIOOBbbourhood of e~ of Its poinU and eeeee u_....

AI for prapmy(4). if x E Int A lMn , obviou, ty. }lE A andx'l"(.X' A )' . ConVUK.Iy. if Jl E A and Jl (.X , "' r then th~ il; :anei,lIbou rtlood 0 ",") CA . therefo re,X l' lai A .The vm fica1ion of propert y (S) is k ft to !.bl;: rc.clu. _

lbe~ ltI1 (.X' A)has to be c:on.sidc~ quite oflt:n II Isaflcd 1M u.tcrior ofthe lei A IUIddmotfd by u t A .

~l.Showlhat A " X 'edA .3. The Bound ary or a Set. The fo11owinS important conceP'S.llfC those0'boulld ary po int an d the boundary of. Jet A . They ..-c~ecnu.nd Iea,'c the ot hen as Q

L" . 2. Getlcfal Topolou"

E:nrcises.r . Ln U bc opm In X IIfIdA - au. Show lha t aA '" A . Prove the converse state-""".,.. Let Y be a 1IIbs.puc of a lopoklgkal spac:cX , .andA a sub$et in Y. Dmo~ thebounduy oftbe idA in Y by ay4 . and the bourw:!.an'orA in X by;)A . Verir, thaI itis tlOt ahfayS lnIe Ihal cJJ"'l .. (aA ) n Y . Give some examples.

7. OPERATIONS OVER SETS IN METRIC SPACES.SPHERES AND BALLS. COMPLETENESS

1. Operations Over Sets in Metric Spaces. Here we consider Ii'll' COII-~s Sludied in the previous stet ion as they apply 10 metric liPKu. RemernbCr thai11Kbase for the lOPOklsY in fJ.I. p) COlllUts ofall poIIible balls D,t;co>. where" > 0isthe radius. md"'l1 is wanln orthe balL The P'lCtric ,. makes it possible for U$ to$peak of aoD1ltflCnt scqumccs in M ($CIl: Sec. 2, 01. I). We can CllPfcK A. A ' .Int A . CIA in Ihea terms Ihus:

(a) the cODditioa x Ln.t A is cqU!v. ltnlto the ball Dc(ir). for aC'elUin e > O. be-in& contained ..molly in A: this folo\WS f, om the 6f:rmi(ion of th e marie topoJoay

,~;(b) the condition x EA ' is equivalent to the ctistencc of sequence "'.1con-

\'l:l",ent to Jr. where a A , a.. ~ If .In fact . if Jr GA ' the n for anY "1 > 0, there is an clement " I in A such that

IIle D,. ,,"), III '" x . Lei 0 < "1 < p (.c, a ll . then alain there is an element"1 E D, (xl. a1 *' Jr , etc. Th us, the sequ ences ir..Jand 1a..1C A (ire constructed suchthtp';... Jr) < "., ". - 0, a. '" If, Le . a.. - Jr.

Conversely. lei there emt a sequence a.. - s , where a. oil x, a.. Ii A . Then forany nriahbou' bood 0(:0") o f Ihe point .... there W SI a b. 1I D.,,") C Q(r) ud N (e)SIKh th.l p {a x) < e fnf It ;l= N(e) . H mee G" 0 ,,") w~ It ;l N (c) :lnd G. *' Jr,. hich completes lbc proof.

Th e doerlD itioa o f . limit poinl in tmns o f KCI_ cnn~&Cft1 to it livu aboeisa1....ys used in anal ysis as lbc de flnitioo nf a limit po int or. set;

(c) th e condit iOn tha i a itt A is dowd Implies . just like lOt to polngkal JpK e,lhaol A conta!ru lllJ its llmi l poi nls.. This condition n. eqllivak'ntlo the fac1 Ihal th econdition z. A fol ln.. from the emlen c:enf a scqucnoe \ll.l C A convusellt 10 ....III face. th e condition that A is dosed is equi valent . for eumtlle , to lhe con di tionlb. l A C A (ICe 5: . $) which Is equiv alent to the previous $IDtemcnl ;

(d) llle con dilion ;It 11 llA is eq uivalent to Dr "" ) n A * 0 and D , (K)n (X 'A ) 'fJ 0 for any , > O. t.e.an)' ball wilh cent re althe poinl'" will ' JCOOp'OUI the po inu of A I.fId X ' A . Th is st.tement is ob~iOlll .

We ar e also liv inl an equival ent definition which Is n ltm used in anal ysis;(e) Ihe OOIlditiofl x E aA is cqui'flllenllO the uistrncc of a Jocquenr:e ~~J . X ' A

(OIIYCf'1lCf1.t tox. and to the exlSlenoc or. It:quencc 1a,,1C A "OIIVCIlIenl to x .In fact. su ppose X li! 'M. Then for . ny" > O. the ~n Dr,,"l 'KOOPS' poi nts OIlt

of both A (i .e. , tbc po int a, )~ X 'A (i.e. th e poinl G; ) . AswImin a!hlll. ' .. "". _ O. we obt&in th.. sequences G,. E .4, .~. C X ,A sudl tha i . '. _ x, G;. - x,

"Introd uction co Topo\oJY

Convtndy, ih. - x.Ia.1C A. and/l~ - z. I'r~ 1 C X , A., thmany baUD, (.l") coa -tains both the point " . U>dth e point ,,; fCll' a sv.rrociently IaraCIi .. " (r); lhererart,JUliA .

2. Balls and Spheres in R". We lba.Il invcstip lc the sptlcrc S , 1heopcn 4ise1>" .. I an d the dosed disc1)- ... I in R" .. "TH.EOREM I . The/ allowing tqutllil~s are Wllid : U- .. 1 .. u)i+I) .. (D" '" ' ) .PROOF If llle 'r ay ' [txoJ, 0 " t < +00 , I, co nsidered (i t emanates fro m tbe cen-In o f the ball, the poun o , and passes t llro u&b the poinl Xotii b" .. 1' %0 .. 0), thenlbe poinlSxi .. Ie ; 1 xoOfthis ray Imd loxOand lie ill D" " l (vuify th is by ou.inI

lhc metric on R" .. I ) , aDdthe points h '" ~ %0also Jie ill D" ... I and I.md to :mo.Therefore, (D" " ,)' ::l D" .. " On the ot her hand, (0" .. I) c b" .. I (JIere(L)i'TT) is lbc lopolocical doIure of the ball D" " I). In fact . it x. - "Xl ii D" .. I . i ,e ., Ify e (D" .. 'r tllen

p (Y, O) " p()' . x. ) .. p(x. , 0) < p(Y ,.ll"... ) + J._ hmce by taklna: lnto accounli llal p(y . Xl> - 0 as Ie - _ . we ban p (jo. 0) " i,i.e.Y E D"" I.

After eombinina: the Indl.lslon malions Ihu we b vc obtained wilh Ihe eviclmtrelation (d' .. 'r c~, we have

u+ I c w .. 'r c (D". I)C 1Y'+ "wheGl% the I tt.teme nc o f tbe theo~m readily foUowI . THEOREM l .~$phtn Is Ih~ boundary Qj o boll : S - .. a(D" + ' 1) .

' - 1PROOf. IA. .1'05 S" (S" ,p. 0l). Then XI< .. - . - .1'0 E lY' '' I, and the tequem:e~tJ u.e., with radillS

0 . 2. o.ncnoJ TOPO!ou"

r > 0 and orllt.~ at Ih" po illt x~ by the eqlluunD,~~ ,., ll' IE M . /I (x. x~ " rl. S, tlr'ol .. 1M' e M. /Itlr' , xol .. rJ.

Not" tl\1I1D,tlr'ol. S,tlr'ol are eIosed XU IDM . l.ll faet , i ( ll'.J E D,tlr'ol and ]C. - ?...

p(E-.. ?} " Ptlr'o-:r.) + Ptlr' ?) " r + Ptlr' ?).YoiImoc pl;co-?} " ' , Le.?eD,tlr'~; S, = M.SI(xol .. M ' lxol and (01(}CO C D 1(}Co) ,Fu.:1.humOfC. (D .(X0l) D I,""ol. $1'""0> ~ ,)Dl(x~ " 0 .Fillll Uy. when r > l,we have

D,tlr'o> = D, ,""ol - M . S,u~ " 0:mOfCO"et".m) '" l'J,Vol *" (D,

"In lr od uetion IIIT OpOlo&y

i.e . for any t > O. there is an integer N {e) such that I x" ... m - xnI 0:: t as soen 111I ;;a N(t ),m ~ I.

If x" !. xoiD W . p ) , then lx,,1ean be calIil)' st.,, ""n to be fundamental 8$ i. t ruein the case of R I, l.e., for any e > 0, there is N(c} such lhat

P~,,+ m , x,, ) t:;II: . ll " N(c). m ;;a I. (I )However. Ihe conven e i. nOI always true .

DEfiNITION I . The space lM . p ) in which Cauchy' s criterion hokl ' true Il.e., anyfundament al sequenc e h M a limit) is eaIled a clImp/ere spare.EXMll"UlS2. Let M = Q c R I be the set of rational num ber s in R I. Th is metric space i . notcom plete since there W s! seq\lcnc~ of ralionalll\lmbcrs convergen t lo an irra tionalnumber (i. c . , funda mental, but havina no limit in Q ).3. The. spa

ClI. '2. General Topolou n

THEO REM I . u lf: X - Y W Il COllI /nuoUSmQpp/II . T1I' loJJow",. Pf oPNtiullneqll fwlll!lIl : ( I) I is COIItifl llOUS; (2) f or Il lly A e x. /(A ) C nil); (J) frK Il fl )'A C Y._h~: rl (A) C rtIA ).p JOOf' We sh an prove _ impa tKxls. (1) .. (2); We rondudc from the defmition of ~Oftlinuity thai the K1/- lmJi is dosoed in Y ",d contains A . "ThercfOl"e .A C r Iqvm .!"'hetK'C / (Al C f (A). (2) .. (I): Obviously. it follo ws from (2)thatA C r IV(A fot any 1 ' Choosin&~t(F). whtre F is an arbitrary c1os edICt in Y.We obtain thalr (F) C r IVV- (1')) - r l(F). "ThercfDTe.r I( F)is~Io$ed fOt any closonrl (Al C.c ; IA ). (J) .. ( I) ; For a closed A, the chain of itIclurion relat ionsr l(A) ~ r (A ) ~ r 'IA) follo ws Item (J) . Whencc r l (A ) i. dosed , a nd thcmapplna / Is therefore-eont inuolU. i

On the analogy o f the deOnltion of the co ntinui ty o f a mappln8 in a mettlcsp.~e . co ntln \ious mappi ngs of topological spaces can be defined as bein a con -tinuo\is at every point by intTod ucina the noti on o f continui ty o f a TRap ping a t apalm in a topoloJ,ical space .

DEFINITIO:>

"l "trod~ction (0 TQpology

(ii) Given a topological spaceX, a $el Y and a mappinlf : X - Y. Equip Y witha topology so as 10 make ! conlinecus.

(iii) Gire" a topological space Y, a w:t X and a mapping! : X _ Y . Equip Xwith a to poloiY so as to mue] conlimmlls.

Problem (i) has already been and mappinp. Tosolve it, Ira information about X. Y, and ! ;' rtquired.

Problem (ii) ClI.Jl be tolved withollt any additionalauumptkl ,. Let iUl = .. be atopoiolY on X . W. endow Y wilh a topoloJj' by ea.lIing upen in Y those and onlythO$e sets V C Y whooe inverse imagcs ! ' (V) .. U arc open in X (includin g theempty inverse image) . It is not dirn

01. 2. Qc,ncral Top01olY"

aiM. Since! .. , .. .r

so Inlr od uClJo n 10 TGPOloJy

l i5 open, we find l h.at the $Ct/

eh . Z. Ge".... IT o pol"IY"

Exercises.2 . prove that the two-dimensional10rus r 2 is homeomor phic to lhe l>rodu cts' x a'.3 . Pro" e that the space st x R I is homeomorphic to the circu lar cylinder.

Consider the project ionsp \ : X x Y - X , (I:, y j -x; P2 ; X X Y - Y. '-". y) -y .

THEOREM I . The mappings PI' P2 are comjnuous in the producl IOp% gy.MoreOYl!r, this is 1M weakesl topoloBY in wh Ich P I endPz arc continuous.PROOF. We show that PI is con tinuou s. Let U" be a SC t from th e base for X . It suf-Iiees to show that P i"I(V..) is o pen. Since the spac e Y ca n be represented Ill; lheunion U V, o f all the sets of the base

P i"t (U,,) _ U" x Y _ V" x U V,, "" U (U" x Va)

and therefo re P i" I(U..) is o pen in X x Y_ The continuit y of P2 is vertfled in thesame way.

Let us ~erlfy the second sta teme nt of the theorem. For PI 10 be continuous, it isnecessary that the sets p i"I(V..) '" U.. x Y should be open. Fo r Pz to be con-t.lnuous. th e scu X x VII "" p i l(ya) sho uld be open. Hence for bWe shall describe il in more detail . TIle subbase for tile Tlboecv topo logy can most"sily be characterized as Ihe collection of all th e possible set s in the produ ct ~~"X..Which have th e form 8 " ... f% : x(aOJ C V.. I. Where U" is an atbitrary element ofIhe ba se for lhe spaceoX~ and a oE A isoabo arbitrah, . It i$ easy to see tha t

B~ _ P;; I (U" ). Thus, fo? a certa tn "'0' the sets (B..J fOf"m the weake n topologyon~ n X"oin wh'kh the p.ojcclion P", is continuous. :rIierefofe. having declar ed the

" ~ [8",l.. ~ A 10 be Ihe subbase , we obl ain the weakes ttopoloSY in which all the pro-jcaions p,, ' a rC CQ ntinuous.

Hence, "the ba~ for Ihe TIl'onov topology COn.iSIS of sets of the fonnU .. p ; ,I CU" I) n p ;.I(U" I) n .. . n p;~I(U...'.

"Intr od uClio" 10Topolosy

""hu e" , . . " "11 1$ a n arb itra ry (inil" $C1o f "lemen IS lO A , and V.. ;~ an arllilta rydement of Ihe base IIIX" . In othe r words, an open .seto f the base i~ a .set of fun' -tl ans I

x u..~

THEOREM ~. For QnY " o A , Ih" projtio" P"o : nx.. - X.." ;S Q~fl/l..ousondopt:n m ll/Jpml.

PROOf. The continuity ofP" has already been verified. Since IhcP"oimage of a setfrom Ihe base for the IOpo!tgy i, open , the image of any open ~t IS aha o pen E.xercisJu.4. Silo... that the ba se for the Tihonov topology ;5 formed by all possible .sen or lheform

(SIIt h sets are of1m re ferred 10 as cylinders) .S . Verify Ihlll R" " .RI )( , . , >( R I, D!::$Cribe Iht base and the subbase for the

Tihonov topology a ll R D ,6", Verify thai tile ,,- dimensional cube I" in R " can be repr esented nsr .. J x xJ,wherc/ =- IO,I ].

7" , Consider the n-dlmtnsionaltoru s "" .. 5 1 X . .. X 51, (there are n facto rs)and eesertbe the subbase and base for its topo logy , 2. Continuous Mappings into the Product of Spaces. We in-vestigate the lIlappin ss / : X - ~~AX" from a topological space X inro a product.weean C()I'l sider the components/ " : X - X" and / .. '" pJ of lhe ma ppmg/ . Con-versely, If a ~t of lIlapping$' !to : X - Xo ' cre A l is given, then the mappin,g/ : X - ~~AX O is determined uniquely.

Thus, there extas a bljealon between the set o r map pings/ : X - n X" andthe family of the ~tS of m"ppin S5 !tol.. ~ A ' " ...THEOREM 3. VI mapping/ iscontinllous) ." (Ihe mupplng/" ls ronrinuous/OT NChere A ).PIIOOF. u t all/" be cominuous. In order to sho w that/ is continuous, it sufficc$ toshow th.atr JIU) is open in X for any U from the bale for n X"' Let

. "U ( n Xo ) )( U" l X , . x Uttt ,\., " ~ ,.. ' ''k

thenr l(U) " lx E X : /", ~l .. X.. , er '#. 01 ' " . ,o* J",~) E' U",' 1 = 1, 2. , . " k)

n r.;1(X,.) n /;/ CU..,)n , . n/;/(u..,).. .. " I' . ...

Ch. 2. G~MuJTopoIOSY""'''~"' V, ... .t;, I (V..) is an open $et In X owing to th~ co nt inuiry aU.." Th~refore,r IIV) is open in X. The proo f o f the eo nver$ . ) , an d I Ix;. the restricuon of110 X"". The diagram~ 1/xa, '"' It.~ ..,/'

Jf,,:,~can be:nal ura lly completed 10 a commutative One by the product o f twO mapplnlPbee th e dotled arrow) . We :) (x ,!(r:

(2) if is co ntinuo us) .. (/ is co ntinuous);(3) if is continuo us) .. (f'f i. dosed) .

.. lmroduClJon lo TopoJOJY

1) .R~qu ivaten l poin l.iin X(x ~ y)ean be rorn blned into a pair (x ,yleX x X ;denote ure set of ..u such pairs by R ex x x . Sho w that (I) if X / R is Ha llsdo rH,tben the SCI.R is clOllW; (2) if th e projecti on ". ; X - X IR u open and the K t R Isclosed . lhen X IR Is Hausdorff.14 , Show thaI the produ ct of Hausdorff spaces Is eisc a Hausdorff space .lSQ. Show thaI 8, space X is Hausdo d f If and only If the diag on al ~ = \(.>', x) is

Ch. 2. General Topol08Y as

An 8 :t 0 1U1d A andBare non~mply. Th~rerore,A C X '8andii C X 'A ,i.e., A _ A, jj c: B, which implies the dosedness of A and B . The c;ondltions ofDermition 3 arc Ihus fulfilled .

(2) Convcrsely, kl X be dlsconneeted in the sense o f Definition 3. ThenX .. A U B,A, B are ncnempry and open,A n B .. 0 .lt is obviou s Utat A andB are closed. Hence, An B = 0 , since A co .4;jj n A = 0 becauseB .. B.

Th~ foll owing theorem produc~. an impQr1.ant ~"ample of .. co nnected space.THEOREM I. Allyugrrunt ID, b] of/Ile number fine R 1 is cDnnected.PROOF. QlI\sider a topological space: X = la, b) equip!lCd with me topology fromR1 . Assume that X is disconnmed;X - U U V, u n v '" 0, where U , Van:nonemp ty aPd open (and simultan eo usly dosed). LeI a e U. W e will consid~r line

~gmenls ill, xl where x e (It, b ) .When x is near to

.. Inl,odllCl ;on 10 Topol""

lHEOREM l. wI : X - y bt (I cOlltllluOIII mDPp;",. 1/X Ir ronnltd then/(X)ir eo'lIrtrt ill Y.

PROOF.~mc IhCCQntrary , i.e .,f(X) .. VI U VI' where V I n VI ~ 0, V V,bcina opcalnf(X). V. ~ 0 aDd VI .. 0 . Thu/(X)Uopt'n implies tha I there a -1$1 two 5eU U and v 1ha1 are ope n in. Yand NdJ thai un /cX} _ V ,.V nf(X) .. V, _But ,dearlyJ(X) ~ V. U VI .. X ""r'W,) Ur 1cYJis!be11IlM>n of llOnlhpt1,dbjoint w bseu. SJnccr IW I) .. r 1(U),r ' ( VI ) .. r I(Y)(why1) Indr I (U), r'cn are opt'll Ids (due to the eontiDuily of}), the decom-po$Mion X .. r ICU.) u r lCVj ) tsCOllU'at)'l o the~ ofX . Eurti# J" . (IJ Show tIw the~ r/o r, oonLinuous mappmc!or. Jnnee'ltdI:J)IllZ is Cl:XUItcd .

(b) Hcoce daSuee the thCOrall that the nume rical eon tinlIOUS funetion I : [0,II - If IW. ZCft) f ; / (0 .. 0 in tile intcfYIJ (0. I ) IfJ(OJJ(I) < O.

Stat ana-c l (b) of~dse J is a varia.lll of the dauical Boluno thcoran Pl"ovui.in anaJ,ysis . To the Bollano theorem, I more peraI lnlermec1iatc-val\K lhrorml lsreW ed, viJ;. , If . numeriea1 fUftetion I fIt) is tontillllOU. on aIi~ ,,", 61.

J {fI ) "/UJ), and nUlllbe:r C is inc:ludcd lJcotwa:n the Ilwn bers / (D) and/(b), th(:llthe re ClII$U . point C E" (Po 61such thatf(e) .. C.

Thi s theorema1lo fOllows frOIll Tbewcm J . ln fad. th~ intam~I~valllCth~orem is equivaknl 10 the nonemply interseciioll of the , ra ph r, of _ numericalfunclion f (';) with the sU'ai&lll lin~ )I .. C in die plane R 1 wltic:h follows from Iheeonnccr:ednc:sso f the ,,_ph r, and the choice o f the number C.

The Bolzano en d the intcrmedille-value theorelll could h.v~ been provedwilhoul rerol'tlq 10 lhe "aph of the map,plns f . instead, lhe proof could haverelied on the collllccr:edn~ (in the space R ) ofltte lma~f(lI1 , bll an d the propertyof connect ed fClS in R I io contain evee)' Inl mncdiate point tOSether willi and be-tween artY 11110 poinu (prove!) .Extrr:istt 4- . Pr ove that tbe circum fere nce s ' il conncaed

Hl#tf: ConlIIdcrtbo mappiq [0, 1} _ S' livc n by Ibc fonn:u!llcx '" c:os2.-I, ,. dn2.f.

The followina thCOl'm1 is intull ively obvious.

THE OREM 4. A SptIC'r X i..- QJfUIttd if11" )11_ QjIUptJUIu ctVJ iN 'joUud' bJl_CtNftw:t:ttdsa.bW (i-~_. tllq lk iIt 11 COIf,,ttt/ SIl~/).PlClOP Aaumc the ODDltar)'. Then X .. U U Y is th e oorrespondinl decompoAitieD into opnr putl (U n " .. 0). Let Wo e' V aDd VOE " be lWO poinu , an:!L C Xa conneeud let roIItailun& " oand VO' Put V I " v n L and "'I .. ",n L .They ~ open (and llontD'lPlY) s.eu in L , m

Ch. 1. Cene,al Topology"

equatori al sp here Sf' - I, and eac h hemisp here is ccnneesed as it is a co nt inuousUnase DC the disc (see Sec. 1) .

We shall now establls h the following more gen eral crilertOll o f connectedness .THEOREM ,. Giwm a f amily 01 g t.. IA,,1 tllat a~ contrtffl in X fI"d PQil'l'lfiseu~purattld, t ile" C = U A ...isconnted In X .

PR.ooF. ~ume Ihe co ntr ary : let C ". DI'U D l , D I n D l .. 0, lUld D I, D l benonempty and closed. in C . For an arb itrary A ..., the fo llowing cases arise:

{1}A... C DI, roA... C ~ m~nDI*0, A ... n~~0.HoweYICI", ease (3 ) can be excluded d ue 10 th e cnnn cetcdneu DCA " . Hen ce, weh~Ye the sets A ..., C Di, i .. I . 1, but the cl~nC$S of D j in C implic:s that(A" n C) C D/,' _ ..I. 1 . _

ills evident that (A"J n 0 n A 2 = 0 . A ... n (A !'l n C) .. e . an d.by tak-i!!s inlo ace:ount the mcluSlJl ,lations A ....t: c, i = I , 2 . we obtain thatA", n A" _ 0 and A..., n A" a e, whi ch II con trary 10 the usumption. thatA and 'Ji" are not separat ed.

" 'one spedal cllW of spaces satisr~Theorem 4. They ar e termed pai h-connectcdspaces. To descri be them, we teeceuce the concept of pa lh In X .DEFlNtTION 4. Thc co ntinuous mawins s: to , I} - X ,s(O) .. a ,s(l) .. b is calleda ptJth connecting two po in ts ' and b DCa topolo&ical $pUC X .~l'C~ 60 Verify th at the lmqe $( 1) of the Une-scgment I .. [0, I ] is a conntcdset connecting Ihe points a and b.DEFINITION j. A space X ill said to bcPQth..-orlJ1 lffl ifan,. two polDu in it QUI beconnected by pa th.

An exam ple of a palh-conn cettd space may be gival by a closed su rface (5Sec. A).

It roll ow", from Thoorem 4 that a path-connected space Is neceuati.ly eo nncctcd .That the co nverse is not yalid can be demo nstr ated by the followl.ng exam ple . Con-sider the uni on of sets In R l:

X = [(0 , 0 ). ( I, O~ ~ I[G,0),G-' I)] U (0 , I),uu:l.deno te the line-lClP""nt co nnecting the poi nts P an d Q In Rl by {p . QI; X i.1

~onnccted but not path-eo nnccted {the point (0, I) cannot be co nnected by a pathwltlt an)' other po int from Xl.Extrclsu.7- . Verify thac convex sets in R" and the sphere S" ,,, ... I , are path-connCCled.S. PJovethat if A C X i.1 co nn ccted , then an )' B such that A C B C A Is also eon-nected . Qiv e examp les.

FinaUy. we shall co nside r the product o f con nected spaces.THEOReM 6. The f)'Oduit X J( Y o/COlrIWCted Sj}QCIU is contrtw .PROOf. AMume Itle contrary . Let X x y .. U U V be a deccmpcsnkm tmc OIXn

..

(oonempt,.>letS. U n v "" 0 . LeI (.r(loYO> E' U. The sc t.l'o )( Y is hom~onorphic 10Yand. lha'efor c connected ; inlentlna U aI the point {.loo. ')10>. II lies 'lItboUy in U.wtticb followl from the COMK"tod" GS o f U. The leU :x x Y. ye Y. inlcncd~ x Yand lhenforc U. Howeva. bdn&ronnected. they lie MioUy in U. Th us ,

U (X x JI) ,., X x Y C U . TherefOR , V .. 0 . The conl~dil:lion proves the, Htheomn. .........

9- . Pro~ Thron:m 6 for the pTod o.K't o f II conncc leCl spac:es (II > 2).10. P I"OC the mnneetedneu of the TibG/lOY p .odud. n X.. .. Y of collMC1edlipac es X" . ....

H iJIl: Collllcler I M .-t R of tbe l!9lnu or the prod lll:t !hill can be Joi ned 10 a CftUin point byconn~ oct!, aDd vaif)' that R .. Y.J. Connected Compone nts. If a spece II d llCo nnec;lro Ihm it is natural 10alt cmplln decompo se it inlOco nnected pleas. We describe th it decomposition. Letx e X be a point in a lopoloakal $pace X . Consider the lar,cst connected seicon-lainin, the point x ; Lit " UA.... WhCfC all ~Jf ~ co nn eded seu eoIltalniJl& tl\c:point x .~ set Lx Is c1osrd.!in the clo.w~ L",orthe con nected set L" Is ronntedIsee EllCrd_ 2) and hence L., C L.. l.e.,' L~ '" Lr DEf ll'(lTION 6. The X I L" is u Ued the COIIn tl f:tmIpofle'll of a poilU.., in Itopo loPtal spaa: X .

~x, yeX. x ~ y. COnsidathc Ku L~, L., Owill,l lathrir~nnedednas and:~IY, there are t1WO pouibiliti5: either (1) L~ .. 1:, or (2) L~ n l

ClI , 2.

'"Inl ' od l>tliOlf lTopo!oty

PlOOF. Ld g be a co unl1lb\e base for the lopoloJ7 on X . SiDu aDy~lemtnto f tllC'Il\'Uin,a lUJ is lJle lJPion o f litU from 9 , a IIlbfamily C QII be~ our.in .!t,wtr,\dl is It _ collJlU.bk . abo (Oven. X and such lhat eKh clancnl of C is eon

~in some detr\ent o ftbc fam ily lU.l. Then . tlavin. dlosai ror-eacb dement oflbe COVUVII C . Set from lU.l tontainina ;1, We oblairl at most a COWltablc JU~eovmn, of the COVU'in& [U. }.

&elides Ule blue for the lOpO!oo mrroclllCed in Sec . I ,~ II. the importanttncqn ol base for the MiJbboumood 5ysl.em of apoinl xof. topological Jp.Il" X .OFINITION 1. The fam ily IB~)\ or the ndahbourhoods or . point Jl is caIltd lilt~lor tile Mf,hbourlroodsystem of x if lhue is ndlhbourhood from this familyin each nei&hbourbood of the DOint :t.

Th us. the family of all esee nelghbclurboo(b or I point is base for theneighbourhood system o f the polnl .

EJ{,.\Ml' l.Il ~. Let X .. Mbc IImelTlc space. and

B~) .. ( l'*CK) " [ Y ; p(jl , x) < iJf_I base for theIlelghbourllood system of tile point Jl . A sphmcaJ oei&hbo\lrbood anbe maGe a rermtmenl of any nelghboUfboodob . For any sphtrical neipbourhood

ID. (K) .. 11 : P~. y ) < IJ number k can be dIosen sudI llW k < 4:; thai

~(K) C D. (:c).DI!F1Nrno.~ 1. A IJ*lC. X Ii said to satisb tilt fus r r:oIIlI flJbiJir, IDCiom If thencqbbcurtlOOd sy$1elD of M Yof its poinh~. rounwle ba$e .

EXA~u>.4. A mmic JPKC is a first COUDlabk spece .S. Th~ l pace Of continuous fUndlonJ Go I] 58.liIfies th~ rlr$l. co untabiJil )' Worn

DoeI lh~ $p~ C tO, I] Atilf)' tbc lUO n'd coun tabllil)' uIom ?Thai it does fo Uo...,from C IO. I] bdns separabl~ , aIld Wo from Thcorr rn 2 of this scctioa.

1lIc separabilityoft!'le, pace Cia. II follows from theWdc~l f"U'; theorem wblch, tates lhat ltl y eonlm LlOU, function o n the line' Kjmen l [0. I ] can be: unlformly apoPfo.dmated by a polynomilll to any dq.ree o f Dccuru y . Thus, a countable anti(v~rywhcrc dense .set A in CIO. I' consists of lbe set o f Illl polynOlJ\lals fP~(f)J wilh4"lllonlll cod fieienls.

~_ 1- . Veri fY that space satb fyinJ the seco lld co LlntabililY axiom aboSIIusrles the first.

TlW th~ conyersCOIIn lable .lpalle X with the disc:n:le lopolOlY satis rlCS the nn lCOUllllbility uiom. 1n 1at'Y poi1It ~E X pos.s.es5a a base for the neiahbouthoocl1)'Ittlt1 COIlsisIin, o f a linlk nci&bbouthood v _ ~. Bll t Iud! a space does AOIAltilCy Ihe second COWJtabi\ily utom. 1bls follows from liIlddor l theomn andftOm the fad l.hat the coverina formed by Ofle-poinl sets ~. ~ e X , hu no COWl-l.ble lU!xOverilli .

O . 1 0.""..1T opoloay"

Th IlS, Lbe fulfilntmt of the second couftUbitity axiom is . st ron&cr condition on."lopotopc:aI space Lban Is Lbe fir St COWl."'bi lily uiom.2. Properties of Space Separation, SOme imponUI\ lopoJoaka.I prOpel'tit$ are dlaradcrlad by t be scpwMlon uiorns. These uioms mabie IlS 10 rcsux.i"'= cLais of the~ Sludird ill ordn to lft$idc:l' tbdr~ properties.

We 5b.n .sdllDe~mUD MfItlration fIXiom8 T. -T . A~ spacC' X b Ald10 be. TrSJNI" if J.ll uioo'I T,. i _ 0. I , :2 , 3. 4, i5 fulfilled for il .

(To'. At f~ OM oj lIl'I)I ,wo difJ~'It p

"In,,,xl\lct io n 10 Topo los r

point. It is e.u y 10 see that this space i. HausdQrff but not reaulat , because the dis-jo int ~losed 5C1 E ... ('; : II ... I, 2, . .] and lhe zero point are nOl: separated inthe sense of Illc Miom TJ .

'rxese c:umplelldmJ01\Slrate th at the class of rcs ular (and aLw normal) spacn is~nliaUy fl8I't()wer than that of me Hausdorff spaces. How~er. the clan of or.mal spaces ill qulle wldc81Id includes , fo r example. all met ric spaces as will be sbcwnin the nat soclion.Exercises.2", ShOW thai in a Tl-space. for any ",bscr A . the following inclusion relation isfulfi lled : (A " ) ' C A ' ,3 , Verify that a dOSl:

A ", j (A ' ) C/(V' ) C j d ;;' ) C j ( U ' ) = U,

The itnaie Dt a normal lp a

"R'

In O. such thaI

I ", (x~ - .. (xl I < 112"' , x I;' U",(x~ . Le t ro (of the ronn kl2~) be such Ihat,,(-'Ol < ro ,,(x~ andro - 1/2 .. I < ",(.x~ .l! x C U,,/.xol lhen X I;' G(rr). Therefore ..(~ " ro' Fur-tbermcre ,

x EX' G(ro 112N) e x , G(ro - 1/ 2"' ),

Ch 2. Gcn..al Topology

therefo re , o - 1I:r' .. I", . ",and s~ 1.(.%)1 "" ~ 1",(%)1....OOF. We $hall co nst ruct the function'" u the limit o f a certain sequence o f Iunc-ucns. Put ...o = "" and

It is clear that the seu Ao Boare closed and d i. joinl. By the ma jor Uryson lemma.there exists a conlinuous (un ctio n ' 0 ; X - R I such that Igo(.%)1 " ~ and)

t (.%) = (-1'01' if xa AOoo 001' if )C 6 8 0-

Now. we define the funct ion "'I on A by the equality ""1 = "'0 - 8"0' Th e fun ct ion

"' I is therefore continuou s anda l .. s~f 1'1'11"-iao' Simila rly, by in trodu cing th~notation

"Introduction ." TOPOlDi)'

wh mwhen

xeA,J< e B.

a"" I (K), we obtain an estimate

Let x E A, then the partial sum S,,(x) _ t V;") oj- + 1,,(4 by the method ofccnsuuctlcn 01 the funct ions 'l'n .. I (x ) wilYequal \Po(x ) - ",.. (x). and ",..(x) - O.ThacfoTe, 4> (x) = "'o(x) = ""

CII. 2. Qn >Il ') ifeach ekment of If is co ntai ned in some dement of the system ,, ' . The refinementrclatkm introduces II panial o rdering OD the set. Qf aJI covetin p of the "PAce.

Coverings consis ti", of II finite (or count able) number of d ements are said to bejinlle (or fX}unrllb lej, respectively .DEFINITtON 3. A coverlng a of a space X is said to be /OCtllly j/n ;te if each pointX l! X posseS$C$ a neighbourhood which intersec ts with only a finite n umbe r ofelements o f C/. Coverin&S consisti", of open SCls arePilMlcUlarly important and saidto be ope".

There Me many Importan t pr operties o f spaces which are clo~ly related 10 theproperties of open coverings. Hence. the follo wing classes of spaces are singled OUI.DEFtNITtON 4. A. Hausdorff topologicaJ space X is said 10 be (A I) comPQCt. {A ~j/ruI(ly ComfHJCl, tA~poraCOmp

"Int rod uo;1ion 10Topolo$y

fm it" cover ins cannQl. be picked from the covering I(- n, ,,)::' T I" (H owever , co unta ble subcov ering can be chosen from any open covering o f R . 'rnerercre RI isfin ally compact . Prove it.)

Similar reasoning dtmO nSlrates that th e space Rn is also non-ccmpacr , nc rarc any of il.$ unbound ed subset s. Hence , it fo llo ws. in part icular , thai the require-me nt for a co mpact subset in R" to be bounded is a nessaty condition .3. The spa X = R' is paracompaCl . In fact, lei tv ,,) be an open rover;"" of R I ,

..

TbenR' '" U ln ,n + I] . Each line-segmen t (n , n + I ] is 'a litt l,,' c:

01.2. a-.:n l TopoIasy

and n V..,(Y) _ V (y ) are open ilIld di$join t. Thus, W~ hllve shown that a co m-

, . ,

paC\ set X an d a point not In il can be s.eplnlted in a Hausdor ff space by the disj ointJW. hbourhOOds V(X) and V()'). H~DU, it foUowl th ai th.t: compk:menl Y,X ilopen, and tberefort X ill d o5ed . Exna. 2- . Prove that a coO>p.Kt Ipasscsid noncrnpty intc:nedion.

A dual ~alement o f th e dcnnltioo of. lXlfnlMlet lpal;C Is the foUowln. theorem.THEOREM 3. A Hausdorff spote X is tontpGCl if andonly if alty ~nlrwJ sy.sl~m ofits closed SV_ IS

"'"Im rod uc ll on 10 Top

'"nooF. Font estabUsh thai X ii rqular . Let II C Xbc. ~ d OHd sub lCl. xe X , A .SiIlc:e X ii Hal,l$dorif, for any :y e A, lhen a m t br; naghbou rhoods U...(1 ) . U,(x) orthe poiJlU Y. x such that U...(y } n U,(x) .. 0. The system rU...(y)k ~A fOlm S covering of II; since A is compact, il. finllc wbcoverilla " '" lU,,(.v/ll:". lcan be singled 0 1,11. Bca~ UJI(y/) is [!\dudeX ' U, (.l'), U..(Y/ ) c x ,u, (.l') C X ' il'J. We.rllld , hat U U...(y/) C X , !rI.

I I , . I

Bill: V0...0'/) ill closed, lhc ltforlt. x '\. U U...l1/) .. UIt (xl is an openI . I ,

ncigtlbourhood of the point x , The union U V.,(Yt) .. V..(.A ) Is It l'l o pen,. ,

~hboulhood of lhc SCI A in X . It is eviden t that V.,(A) n UA(x) ... 0 . IJId thereaularity of X ha.1lhus heetl proved..

We now ha ve to prove lb . X is norma). Lei SCl5A and 8 be closed in X nd An B = 0 . Then for any po int x e X. then. a i$l.$ an open nti&hbourhoocl U(x)ror which al 1ea5t one of the rdaliol'l$ U (;r;) n A = 0 . U(x) n B - 0 is truebecause X is reruJar. Conslda the coverin& fU(x}}u x of 1M JPaee X ,..jIll nJdlndahboutlloods &ltd~ a rlftitcoo~rin&~U"lr'. 1 from ie. For each U..,.al. kasIonc d lbe rd ations Vo / n A ... 0 , U., n S " il fulfilled. LeI U ' - VU., bethe union o f those $ell for whieh U" 0 A _ 0 , an d I lmilarly , V ' .. UU" ,, 'ii O S _ 0 . lI is e~y to see t hat t he opm wI$ X' U ' , X' V eonl al " A and B ,",l'C'pectinly, and are wljoint. Th us. !be normality of X !w also beenproved . Gl

1. Mappings of Compact Spaces. We~ st lldy .some Important proper-ties of lXIntillUOUJ ma.ppinp of ODmJ)KI spa~,THEOREM 1. LeI X . Y M topolo,bJI spU, X mmpoct. Y HrTIlSdOf'/J. tmdI : X - Y tI conIinUOII.rtM,Ppilll 71fti' the imG~J(x) is tl comp

""THEOREM 9. LAr tlu /}VK1us lMotrnt brfillf"kd,fIIId1M mttppbv/ bijtiw, tllCll is ..~rpIIlnn."'OOF. COnsider lM invcnc mappin.r' ~ Y - X . Show thai it Is contiDlJOIl/J. LetA C X be aD arbitruy dosed subset . Since / b closed mappinc./(,01) _ ~I)-I(.A) is c\o,cd In Y, which Implies the continuity of thell\appinar I

MJuly examp!et o f compllCtspaces an . when rotUt ruttlna ' adeN" SPica.EX..UlPU!4. LetXbe I Hausdorff (1(:1or space of fO lJK" romp=! space Y. Th dl X isco rnp" et sinceIt Is. continuous lmaae (.,.,;th respect lO lhe projection) of . cornpw_.

Colbida". coatinl,lO!lS nwnroe- ' fllll(tiOAj : X - R l on tompllCt lJ*% X.Tbe toUowtn, Wrimlrus tbcoran wbicb plil" an imponatll part in maxheru.tiallNIIysls is valld for it.THEOREM 10. A ll' lXNllUIlHltlZ fimctioflJ : X - R ' on ..~ SINI" X isboundl fUId (lttGbu ib muinnlm (lind minimum) Wlf~.

~00f'. 1n vieIrf of Theorem 1, thucl~) is tOmpael . By Th eorem 2, any c

01. 2. Genual Topoloty ,OJ

pad: . there Qisu .n dement Jt.. III X.. such thai for any of it.l neij:bbowhooclsV.. "" V(x",), the in lerKCtion V.. n NT., .. 0 for any H:, E 'Oe - Co nsider now anelement Jt ... lA"J E X . E.ch of iu neighbourhoods U _ U(Jt) contains tbe clOSllrcof a eetu.in elementary adahbourtlood of the fonn

(V x V x .. . x V x D X ) _ V "'I "' J 1" " '.'. 1' .

_hid'! is in turn the intcnectioD of. (mite nu mber o( Ddabbourboocls oftbc. fonn(Um x n XG ) .. Y. C X. lt iscleat W I Y. interxcts all the SCU N' '';;. sin

.... .."'/ ' /V"'; n N:.; 0 fOT au., . Co nsequently , if"",, ; and thenfore

V. ,.. ...... ... ;.. if..;E;;.,. ,

Hentt, the nci&bbourlloocl U :::> U (x) interxas aDN' E ; . Since V is u1)iltary, wedcdu ce thl tJt t' n N'(lnd therdore JtE n NY)

N> ,."

H~ are some exampla which~monst~te how the eompattneu of .spaoc anquickly W detC'r'llliaed by lbe. Tibonov theo rem.

EXA._ .....'I . Tbe projedive space RP" is com~ct ~USt. ic is the fad-or SpKC of th e sphereSo.I . Th e k nl lpllOl'StIt Z, ill compllCl for the WI'C re8$OIl 4. Compactness in Metric Spaces. Compacl melric $pl=.t arc often cal ledcomptJdll, and COmpi ct wbspaca an: called COmpact.1s of II~rit: splIU.

The prope:rty or COrtlpKtfloCSS in I metric space can be tllprelkd. in tenns of con~"'Ieat tequenas.

DeFlN ITJON7 , A set X oh rncttic:space M is said to be # qVDIlillJlycomptICI if Iftysequeooe of lu demcnu contains I subsequwoe which is con VCfl

,.,. Intr odu ction to TopolOlY

PROO~ . Let Xbe sequentially compact and closed . Then for lilly e > O. there cxisu flnhe set of points A. _ (xi] sueb that balls D.(xt) witb eentru in)tlc and radius l!cover X . In fact, otherwiSl:. fo r $01De t o' mer e arc points x!' x, . . .In' ... in XIUch tlull p(lr". Jt~ +.P) O:ruell thaI flny ~tIn X ofdiameter less than.l lies wholly in a t:erlo;n elemelll of tile ro veing 1U] .berci# 7 Let. metnc space X be: compaet,and / : X - Y a eontinuous mapping.

Pro~ tbat for