Level of Sets on Spheres Citation Sonneborn, Lee Myers (1956) Level of Sets on Spheres. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/rjtw-fw15. https://resolver.caltech.edu/CaltechTHESIS:07012025-154007671 Abstract Let f:S n X I 1 → E 1 be a continuous, real-valued function on S n X I 1 for > 1. Then for every t Ɛ I 1 there is a subset A t X t of the n-sphere S n X t with the following properties: i) f(A t X t) = k t independent of x Ɛ A t . ii) A t X t is connected. iii) (S n X t) - (A t X t) has no component containing more than half the n-dimensional measure of S n X t. iv) For any measure-preserving homeomorphism, g, of S n X t, A t X t contains the image of at least one of its points. (e.g. A t X t contains a pair of antipodal points of S n X t) v) k t varies continuously with t. Further, if g:T 2 E 1 is a continuous real-valued function defined on a torus, then there is a connected, non-contractible subset of T 2 on on which g is constant. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Mathematics and Physics) Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Minor Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Fuller, F. Brock Thesis Committee: Unknown, Unknown Defense Date: 1 January 1956 Record Number: CaltechTHESIS:07012025-154007671 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:07012025-154007671 DOI: 10.7907/rjtw-fw15 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 17510 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 02 Jul 2025 17:09 Last Modified: 02 Jul 2025 17:10 Thesis Files PDF - Final Version See Usage Policy. 5MB Repository Staff Only: item control page

LEVEL SE'I'S ON SHIBRES 'I'hesla by Lee ~yers Sonneborn In Partial Fulfillment of the Rtaquirements for tha :DBgree of Doctor of Philosophy California Inst.i.tute of Technology Pasadena, California 1956 ii ACKH~ JLi'I:OOEHENTS It if.J with the deoprmt appreciation th.:.;.t, I t.hrulk the members of the r/i.'.;lt h·amatics Dspartment of t h.e Ct;,1:l.f'ornia Inst:i.tut~ of Technologv for the mutheEw.t:+cal training they ha~ o.f!'crect :'.'!l'a and for too f-lna.mcia1 aid they ha'\<e made a.vo.ilable to me durinG too :Last f ou.r years. Spacial thanks are due to Profes1S1 or F o B. Fu.Jler, wbose a :trec:t.i on of the reaearch reported here has toen a sou.re@ of inspiration unequaled in '!!fl experience . ... i i i ABSTRACT Sn X 11 of th0 n-sphare Sn X t. i) f(A X t) :::; k ii) A#, X t t Xt Then for evecy n > 1. for i-rlth tbs following proporties : t independent of x f.:. A+ ., ·~ is connected. ~ (,\ X t) has no co:nponent contai~·.ing .:::ore thun hulf the n~:i.nensiomu meusm·G of iv) Sn X t. For any r:ieusure-pre~ervlng horn.00\'.11orph.:'tsm, A X t t g f) of ~r X t, \"I contains the image of at leust one of its point.o ,. co:nta.ins a pa.ir of autipO(J,il. points of v) kt Farther, if var:i.ea continuously ,rlth t. 1 is n continuous real-vulued f't.mction cfotin,sd. g s~ .. E on a torus, then there ia u connect.edil non-contractible submJt of on which g :Ls consta."lt. ,~ '!'.-.;; LEVEL SETS ON SPHERES n raal variables. n+1 sat "'' =L-, and i=l < 1, i , the n-sphrsi:cc ,'.;l:mbodded in , n} :ts the un:l t cube of = or a.re given by [ xix c In, some The meaaure used. on Sn indicated. so that the measure of sn 1. denotes the 100usuro of ti\ set sat A A II F (A) A For all three 0 is the oh. U.nill"J measure normulized. In the c'kirivations to follow, and in each case the symbol is usect, the is either open or closed and is there fora measurable . is the boundary of A The notation ima·gs in a set S and for u function f:X .. Y For ,:.U,;-•t sot f dai'ined on a set Y carries also the implication that f X 'vlith ia contirmm.rn with respect to i:.he relev-.1nt topolo;~· , ·ll>'hich in all cases w:U}. be t<!C t one induced by t he r-1etric . d (a,A) d (a,b) i s tho di.-:itunc.e .trorn, is the dist.anoe from t~:e point a to t.h0 .Jot ~ tc A9 d (P.) b~ :1.s t :-J.8 A. diameter of the s e t ') 'T"", the torus , is gi von by ti':1J p:..~O(:luct. of 'tHo cJ.rcloo 11 Tha spncia 2 ,,, ,..., , ..f1 ] • ' or one-s pnere~. It s um.vers~. ..~ ".., cover:..ng • -~ t• ,10 i space :-.i.s p1•~,no For convenience p :K .,. T ·, the projection map will ~-:iap -t,he unit sqmU"Gl 2 of E ..2 2 'ldth opposite face s identif:i.e6 exactly onc0 onto All spaces considered are locru.ly connected, 3.nd thuo the co~.:.p onents of a.1zy reference as w:U.l the fact that u.n.y opcm oot has u clci:nru,18r ablt1 nu:;:11.,er of component.a. ..:. Let 0 .. bo an open subt.set of (i :::: 1,2, • • • , rd such that each hus complement containing uith one and, obviously• only on0 cori1ponrmt It vill be established that~ under this a3SUI.upt1on, thero is a comporn:mt, A11 of greater thU11 one-half. Any set contain:1.ng such a. component, A 9 u:Ll.1 be ~~a.id to posses5 "property P". 'l'hus it will ·b. .; 0stabliLlhed t,hat if open subset of S11 , then e:i:i. ther LID-~ 1: i) - 0 ho.fj be au open connecrt..ed subeat. of ,:,:n - OP t'roJn ,;;J A or 0 Let _._ o f componem:. 0 O 1.s un property If A is closed 1.:.md connected• A is open and co:nnectad. ii) F(A) = F(Sn - A) iii) L,1 com1ecrted . of a closed sot. ii) iii) The proof of :i.i1) is th'.', culrxlrmtion of tl10 argum1.1.mt. on 0 Then corresponding to each com.ponent 9 0<1 o of .,.. camponentp For i f. j one of the following mu.st holdg o, there L ; u uniquo .,, ') i} Sn ii) ,-::, ii:1.} (Sn Proo.;·: ~n 0 ,..n ,,n T~ .l - ""'1,i C ,:) C ~-i ?:"i .!', ., ~ .; "';.. ;;. .;;. - Tj) (\(SH - T.) ::: ¢ 1 and i am are connected . 0., r-,eing cm1no1;r!:.ad and dir;j oi:nt, J o1 , l iea :i.n a single component of Sn - 0i • 1) Suppose thnt component, i s Ti . Llo s :l.n aome component of Sn - 0 • j Suppose this component is and 2) 'f ho other possibility is tlui.t O_/"\T., ·- ¢. i:,,\ Then 'fl ..,en s "" "-.: ~ .. oj . But Ti is connected and lies in a single component of Sn oj. But H(T.e) > 1/2 and •~.~a (~"- j I\ > 1/2 ind:i.cate that - .J. r/·, Hence und :~r- ure tile co1apon,31nts of o. O' open sots 9 Sn - T. , and is therefore open ~ Lat O' :: component trust be establishine the lsr1ti1a. ' O.:i.C Sn - T.J, and tho o_.,., 1 Xj are the componentis of b0 ontiroly contained in a singlo tho U X.• j=1 01o contained i n it. is the uni. er: o:f \-,har e the .,; :ts eom.ected it rtJ1.rnt X. o J I-ie:nce each - 4 LEf.ft.'fA 3a Suppose a.TH such that - Sn - T, m an.cl. contain:3 beth 'l' . 0 J Aa ~ t here J.s no SUCrl Proof: disjoi nt but 0..re b ot .h which contain by lemma 2 . Let, $11 - L:it .L e be tho un.:i.on ofll ;;.J.1 1-;-, Clearly none of these ir1tert;cc t s e n Pl 1 .• l. T ., i..J s he an urc cornwcti:i.1g ,.. D. jl,. t. ,:, ~J - J.i ., £ Sn ~,., .,,.., m 'i' V · :1 s lies in X,r, which is open C~r K..': con.11ect ed. und t h~:t~efore such t.hat arcwiae connected . p c Sn - T q "" S :must intersect for some F(T ). p c S IIF(rr;. Let q, us doos so.rno ho<Xl also conta:lna u pc>int, z of p .. T by v-ir tue of p be:!.onr, ing to 11 But z c S and 5 11 - lj'-q In intersect, one 1:rust contain the ot her from leruna 2 . - Tq C- eith,3r cnse, however, Sn. point of the open sat T and ua such cannot belong to '"l • Hut then p f:. !1' J. . i s & b,1u.ndtrry L~.1t p .. . TJ'."ri s cont radio- r;, -tion e&tablishes the L~rn111.9. . L.t-"1.·,iMA 41 Each Sn - Ti sets ¾ can 00 mrpresaed as a countable expanding mtlcm of (ioO e ench subset contains ull the preceding) . I:... be expressed as the o~dered uni on of the u i=l c orrtu..i.ned in i t o wt 00 x.. - -k 11 S - Ti chosen as follows: :l.k wt T., J.k fo:c sn .,., is the smallest number :Co:r which ,t,~ m consider ='-1 Ur. • Since J.C L{-- for all -k-1 ., :;;: k t t} m k ::.- • where • /~ow I fr:. 0 For any x i:. X., thera is a mnnJ.lest rmmbrrr K 00 ,.. .::iuppose Sn J , Hhere m· . . =- U Tit o T j CZ J 11 k==1 p, for which t •:nen n.,t u,.,o. Jr 8 iz too 1urge:.,t number far Yhic h r.. a first number i'rom CJ.eurly where r :i.n, < j o Then ti'1ere is ~ conht:i.n3 both Sr. - T p • no·it'r1o r cont,a:i.nea • • ·' in • nor a:1.sJ •• •01.n • t 1s is the gres.tcH1t m.u"lber for Yhich "'.i.P -- I r+1 • Therefore • a Notice also that the above argument :i.s valid for tho cam~ whe1"e there are only a finite nuinber of equal to the last Ik. 11 , and that in this casH, in f'aet, .c '!'his completes the proof of lew.wa 4. Now each Sn - X. is e:lther one of the 1 T. J as the dscraa.sing inters0ct:i.on of a countable number of closed and connected. is connected. ~ By lemma J.8 of page 80 of ~✓:Ucler i• x 1 Then for each and Tj fl .;h:lch 31''8 ,. 1 1.2 I) <'.'n 0 - xi o.re co:nnected 1> whi ch '."It; = S0 - THEOREM 1: U (Sn - T~.) connected . i=1 Sn - 0' Pr~f ' - Suppose A and B ...re disjoint, non-empty, tmd r i,lu.tively o• fl and hencs closed in sn • aet Since each bounclur,r ll each has its bounoury entir0ly i.n Then consider A 1 U B' ru1d B' = B v U Xj e J -· { j IF (X .i ) <; 3} • clo3ed in the closed A' ru:id hua co:n:nected. or ~ntirely in B1 arti clo!Jed o jc.J Hence Sn is not com:.ectod. 'I'his cont radiction ei.'lt.abli~1hes t.11.G theore:0,0 6 0 '!.ti th the property t,hat one-half (i.e. ai ther Proofs Suppose O has prope:c-ty P) . ..,n - or 0 ,J O does not huvo property P. lemw.as 2 9 3, and I,,, and theorem 1 were all derived f:ro.'"l this hypothesis . and contains 0, has a connected compl ement Sn - 0 1 0 1 , wr,:i.eh :1.B open which nm.at therefore rn1t, ,~I\ - .Sn - 0' ----::> - vo.n_,ii" ' £ 0 1 o 'oe l ong t .o a si ng-.e 1 c ouponen..."' ,\t, Hance each component of A rm.wt be contu..1.ned :1.n a ~Jingle component of 01• Sn 01 The components o.f the 01. , ,,n ;, Thus uro the O• n,_ , Xk -i_ · which a:c·o cxptm'.:iint; muons of fo-.r. each of which open sets 'lr1i th measure no greuter than onie-hal.f has 1;,eamlre no greater than onehalf (3). Hance Then for any r.10aeu:re--_rJ:re se:rvinc; homeorw rpl:.c sr,1 point x e, A for wl1'Lch g (x) £: A. They cannot be disjoint. P. sn - i .• -~1-<J,J.•~1,v " ~ t ;')1 <:>-' -..LUU. since .,.., n .,&,,. C B19 s. ., componant. of oi' For impposc they n.re; t.oon B il and, by la.nma 'j , .S~'- ti compommt i;:; closed is open=> Sn is cloHGd. o Bu.t has prope:rt.y P9 ant.· M(A which is i rnpo.J si bl(➔ since A 1 ) > ·i',.,; (('n ,::, l1as propo1~t::r - Bi :: ) P. co.r:i.bin0d 1/2 . ttenc;:, ,. L~ ;,, 1'hllS ;.: (A1 ) > 1l ,:. and B moet. ,. ra.etric spa.co DBFINI'l'ION: c > o, there i.s an opon aet for every ei th.er O or Lt,~Jl,iiA 5: f'roof: w:Hl 'z:: t) cu.lleu " dfo,::ootor uivi:,i.ble 11 i f 1 l( OS K nuch that no co~ilponent, of has point :'let. ci h0.:tete1° c,::reat-er than K - 0 Euclidea.11 1-space, E 1 P c.. is cia!:'leter di vislble. By construction. THEOREM 4s fsS Let 11 ... K dian1etf:lr divisible space. be a continuous £\met.ion, where is any K k c Kp f Them for one ar.iti only one -1 (k) has property P. ~: For each n, the:re is an open set and 0 K - 0 C K n have components of' diameter J.e::n tha.'1. n Sn. open set on Either there is a connected set A n r- 1 (o) n A n vd.t.h 1/n., f'-1(0) n 0 :t\ :b an 0.,. :l.ts comple;neut has proportJr Po < - n or f(An ) <;, K - 0 f.'(A ) C 0 n Thus such l'! contains components of rneas-ure not greatsr than onG=tw.lf. s-st, and nmot belong to a 11in£lC c(x1ponent. of c.iameter suc!1 t.h.at both (f (A ) ) < 1/n. n or 0 Tl 1\ ~ 0 n Let ht..G limit point J3 K = ll property P becauHe the ~:iequence o be tho component of n t ., I . ,. i • 1 1 J. a ov:~) .::-.-. • 0 •Lve:r:;r B .ih hua - 8 - for suppose {.. :iC1 :e , z... ii:j J ::: k1 has ' Thein • for sufficiently large rn. p 1/m < p =---> X 1 i Bm• For mippose property P. n > n , x /, B . n 0 For Then , if is a component or Sn B1 £ Cj 1 B or co = U i=nCi 1 C/\Cj c1 has Jr: £, 1 B ., containing =¢. F\.i.rthermore, tr,1 this Hence 11 by Ha.lmos [31 i> o =lim H(C1 ) ! 1/2. Suppose B H(B ) - Bo )£t 1 1 and. :Fu.rtherr,10~1 " B Denote by c the c001ponm1t, of' j > i, c erg1.mient of leiru:na J t .,. d (k pk) =p -I 0 1 I f (B ' ) = k 0 has property P and n~ Then from the proof of thoorem 3 we see that for soma and for = k. x c Bf\B ' IP f(:id = k ' COROLLARY 11 r gSn .. K be a continuous function wh.::.m~ Let diameter divisible space . Then thore ig a set for some i:; wr:; such that kc Ko ii) £(A)= k iii ) No component of Sn - A there is a point X C A Proof) A C Sn K h~s measure greater t.han one-half o .:ru.ch trud; d") C ii. o This combines theorema 3 ai)d 4o The theorem or Johnson [4] establistwd i) 0 Ji) P iv) of corollary 1 for the case where is tho antipodal map of Sn ,, proof, co.'miw.:nicated to mo privately 1 follows the same lirB<Z as s~ similar theorelllB of Sorgeni'rey [5J o COROLLARY 2: En is not difu"l](Jter di"''lr:;ible fo:c n > 1o - 9 - -- Proofz r : Sn ... En Let 1> n > 1• , be g:;., \-BU by f (x. ,x,-, , o o o ,xn+l ) = 1 ;-_ more than two points and hence none hM property P.. COROLLARY .3s Let f:SnX I 1 ... E1 be a continuou.3 real-valued function Sn X [O" 11. set At X t which has property P 'With respect to the n-sphex-e arid such that Then for each t [.. (0 9 1) i,, there is a connect ed defined on i'(x,t) == rt 11 S .X t is a for all x c. Av., • continuous function of t. Proof: Only the continuity of rt is non-ohviouso = d (x 1 ,x2 ) + It 2 - t 1 I.. Then f being continuou.e on a compact s pace must be uniformly continuous. Gi~r0n d((x1ot1),(¾.t2)) <a=> lr(x2,t2) - .f(x1,t1)I < c. (x , t ) • 1 1 It 2 - t 1 I < [ wt !r(:ic,t ) - f(:x9 t )1 < c 1 2 • J c > Op there Gxi s t~ '1'oon there exists by uni f orm cont.inuitJ'• auch t hat, i'or arbitrary x Thus such thut r... ~. i .s a cont i :rn.10-us function of t. 1 oo continuoua where COROLLARY J.,g L:it apace. C be a compact closed set which s0pat·at13z Let f:Sn X I -+ M f(Sn X 1 ,1 C. C - 2 ccxnponents of H - where H. If and c, -then for some t c r1 11 f{Sn X t) on a set with property Po M is uny m0t r 5.c:: intaraeict.s C - 10 - Then gf W ·;MA 6: ie conti,nuous v.nd satisfies t.hs coudi tions of Corollary 3 • O be an open set in the \mi t n-cuoo Let Then either O or ..n J. o:f ..r 11: • 9 t1 - 0 connects a pair_of opposite faces of !3:00.f: It, suffices to consider 2 E • -f II the uri t sqwJ.rs ir. dimensional cases follow by taking a cross section. n ), 1 o r~ o AJ.l hight1r Tho ler:irna ml]_ bs proved by showing that its falsity implies the fal.sity of co-rolla..-:--,y ;?, above. 0 nor Let O be e.n open set in the unit aqua.v-e in f - 0 1 2~ Reflect connects opposita faces of O about all edges of the squares determined by t he integers U.'1til all its images in the .:1quare Tha accompanying figure illustrates thl3 construction for n = 3. its images be denoted by Let O and 01 • 0' :ia open ur~d its compleoont is closed. Three cases a.""8 possible for each component A of . O and 12 - 00 If rruch that noit.her - 11 - A intersects no faces of 1) I2 . "l'hen no r e.f".Lection of A interr.;ec·ts a line determined b:,, an integer. 2) intersects one face of A face a.n extension of A include reflections of r2 • Then whon ref".Lect,ed about t.ha.t is obtained. A cannot, hoiJever, lle i:n more than 1 X 1 squares in tho tYo adjacent The components which n X n square . A intersects two adjacent faces of I 2 • Then the eorrespond!i.:ng 3) c001ponen-ts in the n X n square lie in a 2 X 2 square . Since each component in the n X n square must come from a single component in the unit squa..""e subcli'lrision 11 the co:m.ponents o:J.' complement must have diameter < 2,y2 • 01 and its r. X l:1 square can now· be The linearly shrunk to the unit square preservtng the <lecom.pos:'.l.tion hJ.to and its complement. ov But, -in the new decorJ:I>OfJi tion of tho unit square the diameter of each component is bounded by bJ2 • The truth of n this for arbitra.I"J n contr adicts corollary 2. For any ·1 1 f:t .. E , n > 1, there exists 1 r c E that connects opposite faces of In • ..... Proof: It again suffice a to consider I'~ o S.:l..nei.'J r_; ' is diarw::iter 1 O C E such that every component of na.nd of E1 - 0 0 haa diume tar <! For every n, there is u n n n 1 component A of .r (0 ) or of f-1(E1 - 0 ) cormecting opposite n n n divisible 9 given n, there is 0 faces of :f . limit point x 1 - -< n- A::c in the proof of. theorem 4, for li!hich f(x} = r o B , the c~11ponent of' the set n containing x 11 contains A n for u.:rbi trar:Uy - 12 00 ... B ~= (')._ Bn ccnnocts opposite faces of large w· n=l as in the proof of theorem ;•• THEOREM 5: For A, B, and f:A -+ E1 ' ft :B spaces, and thut the range cf h A X B ➔ E1 Wl.0. h:A X 3 ➔ E1 h.t1.v.i. nL; t.he proporty is contained in bot.h the rD.ngo of range of g, there exist a c A f(a) com.p1.tct, ;j_rcw:'1.sr::-1 connected topological are f .r,d the <!i • h £.. B such 'that = g(b) =h(u,b). Proof i Since all spaces undar consj_deration a.re compact 9 ma~drau a:nd f, g minim.a of h 0 and h Ym.lst exist. Let 111 = llll~ (h(a,b)), = min (h(a,b)), ucA acA bci:l bci3 f 0 ,,. ::,,0 respectively ... ·1 = mrot f(a), = min f(a), acA acA =min g(b), and a 0 nnd al in = llla."IC g1 bci3 A, and. bci3 b C and of these area, there is a last time too value first subsequent time n g{b) • b1 h 0 in " .!)o Along each is assur:wd und ~ is assumed by the appropriate i\.1.nction. IDt 1 F (t ) , G( t ), 0 ~ t:i. ~ 1 , be paramatrizat,icns reapecti1,ely of the sub2 1 arc joining the points f(a ) 2 =h for -which 0 of .A found above for 'Whlch f (a ) ::: h1' and o:f the sub-arc joining b.,.., und l·• ,:, 3 "'J < .JG~ < 11) g(b.::,) _, 3 I, = h1. For OVOr.'J t..:I. I) 0 ... ... h0 tind 1"'(b .-. and = - - 13 h < f (F(t,. )) < h ol. 1 and h < g (G(t. )) < h o- 1 ti = 0 eqw.uities holding c:n ly fer ¢'(t 111 \:;hI Consider th.-e f'unct:i.on ¢(t1't ) = f(F(t 1 )) - g{G(t 2 ) ). 2 ¢(t ,o) > O for 1 number t, 1 - ti = 1 o and the right , f or 2 1 .. E 0 given by ~ satisfies > o, ¢ (t , 1 ) < O for t 2 ¢(0 9 0) ::: ¢(1 11 1) = 0 11 < 1 o ¢(Opt ) < O for 2 1 ,j1 (r ) connects oppos:Ue ,Jides of In. r, such that r =0 abo\/9 inequalitieo i ndicate thut is t.h0 only .r1 ( OJ .. p there is a connected set K on which which can -r. f{F{t )) == g(G (t )) o 1 2 is defined and continuous on K. For ✓,1 = h1 • Henco, f.'or 30Iill9 . h(F(t ), G(t )) = f(F(t )} = g(G(t )J 2 2 1 1 h > f h < f and on the torus 2 T, and covering spu.ce for p:E 2 -+ 2 T would disconnect t 2 :: o, 1 I) ::; {t1 ,t;) c. K, for otherwise the sets for Ht.den {o,o) fram (1 , 1 ) is the induced iunction on in lC. LJ:lt " Ea;::, the ur..i versal •r2 defined by is the projection function far there is a connected sot ii ::: tl = t2 by constructionp a:r.d for h(F (1) 11 G(1)) ~ f (F(1)) 'I'he to connect , tind furtl".ter that. h(F(t ),G(t )) 1 2 with the 101.'t · hantl 1 '• A S E"'· then on which f 1 is const,u..'1t such that has infinite point-se t uiarr..etcr . Proof': restricted to an opposite faces by le.rm:ia 7 . fl (i' + 1 , ;z } :::: f (x , 1 2 1 connects n X n by n ttt)\>I,Jver 9 f i ;3 doubly periodic wi. th 1 ½) = f {;;71 i 2 + 1 ) 11 o H0:oce » for ;.,J.zy- r: , thcn•e - 14 ..., }r 1s a po:int n J~,rir1t o:n c}no t).f ':.,":"1a t~•j O Si3g'IT"l.(H"~t:~ to a point .-:.rt. di s t.an.co f'rom ri X :r1 ex) = r0 0 f-·i (r 1 x • 'l'hen which r1 0 Denote by 0 ± c.) 0 has .a li,:,1.it point n • ill the disk of radius n, \.Ii th center R :: ( \ R on is a_ com1t1cted. sot oori"f.1ai1dr:e X n,l: 0 n C is constant, and ha.rs dia.H1ote:r ut least tlp t.;'l;"j in t11t3 proof of theororri 4 . Thus t,he component of cor1tairo.ng diameter at least n for every n. For arr;;- connected, clos~d ::rn.hset of equivalent: i) A is not c:ontract5.blo i n ii) Every c-nai ghborhood of A contains a horr,otopically non- 2 trivial cycle of T • iii) p -1 2 (A) £. E contains ~ componcm.t of. infinite point set diameter. Proof: Th::\ proof.' 5.s c:Jclic. f, [l)o f{A*) be such a ccmtr-:i.etio:n. hence. A ii3 2J.so. contractlb1e i:n contrad:ictio:n to i ). .. 15 ii) =:> ii.1.) o 'i'his follow:i iumediately upor, u.ppJ.i.ce.t.ion o:: the proof of theorem 6 to the function which i lJl .:i.°"'rO!ll if A A* elsewhere e after noticing that O o:n 1 p - (A-it) A and d.ista:ncs :La infin:l:te co:rrt.ru.ns a non-trivial cycle. iii) ==> i). The second haaotopy covering theorem is used f:A X I 1 ➔ T""? {(61, page 54). Assume A is contractiblee A where, for oonvenienc®p f(x,1) = x be a contra~tion of f(x,O) = k. Lat k- ba any point of E2 for which p(k) = k. Then f IA -+ r:, , given b1J 0 a homotopy of p f is a. homotopy and 0 is a map such thu·l". -f is Than by the homotopy co·troring theo:remp there •• fiA X r - r0 (x) ::: k 1 .. Ff' for which p f(x,t) = f{x,t) o rl A X 1 i::1 o. haJ.0011orphis:i of For t = 1, then, we have that A. -1 But A is compuct, and the components of p (A) are f (A X 1) and its translates by integers (from all of wh5.ch it is di;:rtinc·~ by virtue o:f rl A x 1 being ono to one ) • Thus p -1 (A;' infinite cO'~ponents, contra..··•y to assumption. COROLI.ARY 5: .,.-1 (r ) .i. 0 Given any7 f:T 2 1 -+ E , there exists contains a non-contractible co.r.-pommto T 0 has no REFERENCES [11 Ho Seifert and Wo Thrclfal.1 11 +J.!.hrbu.ch der. Topologie 0 Ch0lsoap ( 191,,7 ), Chapter VIII o (21 R.Lo W"'llo.erp ToRology of Vianifold.s p ,~ricu.n ~,{a.them.at.:),cru. Society (1949 }, Cr..apters II and IIIo (4] RoD• Joh...'1aon 11 Jr." Bulletin of tha Americ.g1 l1:lthmriatiffi!..}ociety11 v. §2 9 page 36 (1954) 9 abstract. (5) R.H. Sorgenfrey 0 Proceed:lngE! of t.he Am.er ican )~thenlli:t.ical Society !il v. ~• pages 179-181 (1954)0 (6) N. Steenrod 11 ~he 'l'opoloey of FibreJ3undle§, Prinm~ton 11 page 5t';, (1951 ).