regular curve-families filling the plane, ii

REGULAR CURVE-FAMILIES FILLING THE PLANE, II

By WILFRED KAPLAN

Introduction. The present paper is a continuation of a previous one on thesame subject. The numbers of the theorems of I will be distinguished by theprefacing of the numeral I.The problem to be considered here is the classification of regular curve-

families filling the plane. Two families will be regarded as equivalent if one isimage of the other under a homeomorphism of the plane onto itself. It is theenumeration of the classes of equivalent families which is our goal.From I we have the results that the curves of each family F are open curves

tending to infinity in both directions, that F can be pictured as a system withtwo order relations: C C1C3 and C, C, C3 [+, termed a chordal systemCS(F), and that, moreover, F forms a normal chordal system. This last struc-tural feature of the family will serve as the basis of the classification.We shall term F o-equivalent to F. if F is equivalent to F under an orienta-

tion-preserving homeomorphism. The basic theorem of the classification canbe stated as follows (see Part 3 below):THEOREM. F1 is o-equivalent to F, if and only if CS(F) is isomorphic to CS(F).

For any F, CS(F) is a normal chordal system and to every (abstract) normal chordalsystem E corresponds a curve-family F for which CS(F) is isomorphic to E.

By this theorem the "o-equivalence classes" of families can be enumerated,and an enumeration of the equivalence classes can thereby be obtained. Thislatter classification is made more effective by a representation theorem: everynormal chordal system can be represented by an isomorphic set K of non-intersectingchords on a circle. The chordal relations in K are defined in the same way asin F. If we let K’ denote the set of chords which is the image of K underreflection in a (fixed) diameter of the circle, then the final form of classificationis thus" FI is equivalent to F if and only if K1 is isomorpMc either to Ks or to KHereK denotes the set of chords corresponding to F (i 1, 2). The effective-ness of this classification lies in the fact that the isomorphism of two sets ofchords is determined solely by the order of end-points on the circumference.

In Part 1 it will be established that to every abstract normal chordal system Ecorresponds a curve-family F which generates it. The normal subdivision of Einto sets ,(V) V (J O(V), where (V) is half-parallel, is used. The course

Received by the Editors of the Annals of Mathematics, February 21, 1940, acceptedby them and later transferred to this Journal. The material in this paper and the pre-ceding one was presented to the American Mathematical Society, September 5, 1939.

See Kaplan, Regular curve-families filling the plane, I, this Journal, vol. 7(1940), pp.154-185. A bibliography is given in that paper.

11

12 WILFRED KAPLAN

of procedure is thus: in 1.1 and 1.2 the problem of how to verify that twonormal chordal systems E and E’ are isomorphic is studied, and that problem isreduced to the comparison of only two types of triples in each X(V) and X(V’)in 1.3 the plan of construction of the family F is described; in 1.4 and 1.5 acurve-family is constructed to represent each X(V) of E and in 1.6 the regularityof each such family is established; in 1.7 and 1.8 these families are piecedtogether to yield a regular curve-family F filling the interior of a circle; in 1.9the criteria of 1.1 and 1.2 are applied to show that F is isomorphic to E.

In Part 2 it will be established that two curve-families F1 and F. generateisomorphic chordal systems if and only if they are o-equivalent" 2.1 is intro-ductory; 2.2 establishes a homeomorphism between the sets O(V) and O(V)by means of maps on parallel lines; in 2.3 and 2.4 "isolating curves"and /2,, are found which cut off suitable neighborhoods o, and o,. of thecurves C,10 and C. in O(V) and O(V) respectively; in 2.5 the o,k andand co, are adjusted in preparation for a homeomorphism; in 2.6 the home-omorphism of O(V) on O(V) is extended, by means of the sets o. and c0.k,

to the boundaries, yielding a map of X(VI) on X(V); this is then extended togive an o-homeomorphism of F1 on F2, as desired.

In Part 3 these results will be used to give the classification as described above.In the Appendix there will be a brief discussion of the application of the

classification method to families with many singularities.

1. Regular curve-families corresponding to normal chordal systems

1.1. Reduction of problem to case of V [J O(V). In this section we shallestablish the theorem that to every normal chordal system E corresponds aregular curve-family F filling the plane and such that CS(F) is isomorphic to E.The precise construction of the family F will be given below. However, inorder to establish the isomorphism of CS(F) and E, it will be necessary to verifythat for every triple C, C, C in F the same chordal relation holds as for thecorresponding triple in E. Our first step will be to anticipate this difficulty andto tke advantage of the normal subdivision to show that the verification needbe made only for restricted class of triples. (See Theorems 1, 4 below.)We shll let E be a fixed bstract normal chordal system, with normal sub-

division by sets V of E0 a [J t(a) and by sets V* of E a [J ti*(a) as in I,3.3.LEMMA 1. If V is a non-void subset of E for which (a) can be determined as a

single-valued function for all a in V in such a way that

(1) [a [J *(a)].[a’ (J ti*(a’)] 0

for every pair a, a’ of distinct elements in V, then V is cyclic.

Proof. If a’[ala" for a, a’, a" in V, then a" (a), a’ *(a) for propernaming of a" and a’. This contradicts (1). Hence a’, a, a" for every triplein V nd V is cyclic.

REGULAR CURVE-FAMILIES FILLING THE PLANE 13

LEMMA 2. Let V1, V., V, be cyclic subsets of E with O(V) determineduniquely (i 1, 2,..., p) (p >- 2). If h(V) and ,(V+I) are adjacent withcommon element tc for i 1, 2, p 1, and if the lc are distinct, then the

p p-1 p

set V V tc is either void, in which case h(V) E, or else V isi=l =I

cyclic and

p p-1 p

o(v) o(v,) 0 ,, x(v) x(v;).

By Theorem 1.27

(1) [a. U ti*(a.)].[a., U *(a.,)] O, j # j’,

(2) [b U ti*(b)].[b, U *(b,)] 0, l’,

(3) ai *(a) (k),

(4) [b, U *(b,)].[(k) U k] 0.

From (3) and (4) follows

(5) [a 0 *(a)].[b 0 *(b)] 0.

(1), (2) and (5) imply by Lemma 1 that (V1 U V) k is cyclic and

[a(). [I a(a)] U [a*().H a(b,)] U ,0(V1) U e(V) U

unless (V U V:) 1 has only one element a, e.g., in V1. In that case

((V U V) kl) O(a) a(a) (a(a).a(kl)) U a*(kl) U k 0(v) O o(v:) U k

if we define 0(a) as a(a).Finally we have

X(V) a U b, U O(V1) U O(V) U X(V) U X(V).

Thus the theorem holds when V is non-void and p 2. If p 2 bu V is void,then V V , X(V1) U a(), x(V) U a*(l) and X(V) Uh(V:) E by Theorem 1.26. Thus, in either case, the theorem holds for p 2.

Proof. Suppose that p 2 and that V is not void. LetV1 kl(Ja.,V2 /c [J b, with 0(VI) a(]Cl)’H a(a.) and O(V2) a*(kl).II a(b).

WILFRED KAPLAN

Now suppose the theorem established for p r. Then let p r -t- 1. Let

W V /c and suppose that W is no void. We ghen have

o(w) o(v,)u E ,.Further, since the h(V) re adjacent,

0(V1) C (kl), 0(V2) C $(kl)

o(v.) *(_) (), o(v,+x) *()

for proper choices of 6(k). We hence have

(1) () (,); z (), , (,), ..., ,_ (,),

whence O(W) (k), while O(V+) 6*(k). Thus X(W) and X(V+) aredjacent. Applying the theorem for the case p 2, we obtain that the setV (W U V,+) k, is either void, in which ease

x() x(+) x() x(+) x(),

or else eyelie and

o( o(w u o(+0 o( u .Hence the theorem holds for p r q- 1 when W is not void. If W were void,then Vr kr_l, whence kr k_l, contrary to the assumption that the k aredistinct. Therefore this case cannot arise. By induction we now concludethat the theorem holds for all p.

LEMMA 3. Under the assumptions of Lemma 2, the chordal relations for eachp

triple a, b, c in the set (Vi) are determined by the chordal relations of triples

in each of )(V1), h(V.), h(Vp).

Proof. Suppose p 2. Then the only case requiring discussion is when, forexample, a and b are in h(V1), c is in (V) and c kl If a k, then b C (k),cC*(ki),whenceblkslcandblalc. Ifaki,bks,thenwehaveboth(1) b kl c and (2) a kl C. By Axiom 3.4 a ks b is not true. If (3) ]1 a b,then (1) and (3) give, by Theorem 1.17, a b c. If (4) ks b a, then (2) and (4)give, by Theorem 1.17, c ]bla. If (5) [ks, a, b ], then (1) and (5) give, byAxiom 3.2, [a, c, b] [ki, a, b]. Hence in all cases the relation for a, b, c isdetermined by that for a, kl, b in h(V). Thus the theorem holds for p 2.


Suppose now that the theorem has been established for p r, and set

W V /c as above. Then, since X(W) is adjaeeng go X(V+), he

chordal relagions in X(V) ),(W) U X(V/) are degermined by ghose in X(W)and X(V+) individually, hence in X(V), X(V), ,X(V+) individually.Thus ghe gheorem holds for p r + 1 and, by induction, for all p.

LEMMA 4. The chordal relations in Eo are determined by those in the setsindividually.

Proof. Let al, a., a3 be three distinct elements of E0, not all in the sameh(V,). Let ai c (V,) (i 1, 2, 3). Let a0 be the largest sequence such thatwe have

o2 Oo #1 # # S >--_ O

Suppose first that r > 0, s > 0, > 0. Then the possible situations areexemplified by either h # , pl pi or hi tl 1 # pl 1 pl. In eachof these two cases, let a4 be the largest sequence such that

O2 O4 ml m2 ma, z > 0

c4 oo ,Ul ,u2 ,Uu, u > O.

Then, by the normality of E, the sets

W1 V4,ml,m2,...,m Vow2,

W2 V ,,,.2,...,._

W.+ V.,

W.+ V.,.

Wa+T+I Va4 ,nl ,nS ,n Vq

satisfy the conditions of Lemma 3. Hence the chordal relations in },(X)o’-+-’r-+

X(W) are deermined by ghose in a eergain se of X(V) individually.

16 WILFRED KAPLAN

Similarly the sets X1 and

X2 V,o,,l a,2,...a,-l

X3 V,o,l,2....,u_

X VoeO,iI1,

X,+I V,,

X,,+ V,,o.Xl

X++ Vo,,,,..., V,

satisfy the conditions of Lemma 3. Hence the chordal relations in ,(Y)u+r+l

(Xq) are determined by those in sets (X), (X.), ),(X+r+) indi-

vidually, hence in the sets X(V) individually. Since X(Y) includes ),(V)

and hence a, a, a3, the theorem is established for r > 0, s > 0, > 0. Ifone of these indices 0, we deduce the same result by a simplification of theabove method.

THEOREM 1. Let E be the normal chordal system considered above. Let E’ bea second normal chordal system, subdivided normally by sets V’, for , C A, and bysets V* for , in A*, with

X(V’,) E a [J it(a), X(V’,*) E* al [J i*(a).

Let f(a) a’ be a one-to-one transformation of E on E’ such that f is an iso-morphism of each ),(V,) on X(V’,), of each h(V*,) on X(V*). Then E is isomorphicto

Proof. Since f(),(V,)) h(V’,) for all a, necessarily f(c,) c, and similarlyf(c*) c, By Lemma 4 the chordal relations in E are determined by those ofthe h(V,) individually (and the way in which they are adjacent). Similarly thosein E0* are determined by those of the X(V*,). E0 and E0* are both sets ),(V) towhich Lemma 3 can be applied. Hence the chordal relations in E E0 LI E0*are determined by those in E0 and E0*, hence ultimately by those in the h(V,)and ,(V*.). The same condition holds for E’ relative to the V’, and V’,*. Butthe ranges of a are the same in both E0 and E and in E and E*, and f is an

’, f(c*,) c,isomorphism on each (V) and X(V.*) with f(c.) c. Hencethe chordal relations for triples in E are the same as those for the correspondingtriples in E’. Thus E is isomorphic to E’. This proves the theorem.We shall apply this theorem below after the curve-family F corresponding to E

has been constructed. By means of it we are able to determine, only by com-paring the sets (V.) with the corresponding sets (V’.) in E’ CS(F), that Eis isomorphic to E’.


1.2. Simplification of the problem in a k(Va). We express k(Va) as

ca.k (J d7 (with d ca) as in I, 4.1.Notation. V+ will denote the set of all elements ca.k such that c, ca.k, d7

for some d V will denote the set of all c.k such that lea, c.k, d ]- forsome d.THEOREM 2. Va. V O. V+a V Ya ca.

Proof. If lc, c,.k, dt ] and c,, c,., dt +/-, with tl < h, then cawhence by Axiom 3.2 [ca, ca.k, dt] [c,, c,.k, dt"]. Hence V+,. V 0. Thesecond equation follows from condition (6) of normality. The theorem thusfollows.

THEOREM 3. The definition ca.k < ca.," for ca, ca.k, ca.," + introduces a simpleordering in V+,. The definition ca.k < ca.," for ca, ca.k, ca.," [- introduces asimple ordering in V-. Further ca.k < ca.t < Ca.,n implies c,.k, ca.t, ca.,"

+ forc,.k, c,.t, ca.,. in V+,, implies [c,.k, ca.t, c,., ]-for ca.k, ca.t, ca.,. in V-.

Proof. Take c,.k, ca.,, in V+,. Then ca, c,.k, ca.,. +/-. Hence eitherc,.k < ca.,. or c,.m < ca.k. If c,.k < ca.c, ca.t < ca.,., then c,, c,.k,

Iv,, c,., c,.,. +, whence, by Axiom 3.1, c,, ca.k, c,.m +, i.e., c,.k < ca.,..Further, by Axiom 3.1, c,.k, ca.t, c,.,. +. The case of V is treated in thesame way.We now renumber the ca.k by allowing k to take negative values in such a way

that the elements of V, are the c,.k for k 1, 2, 3, of V are the c,.k for/c -1, -2, -3, The range of/c may of course be finite, infinite orvoid in each case.

The possible types of triples in k(V,) can be then indicated as follows: (1) ddt, d (2) c.k, ca.t, ca.,. with (a) /, and m > 0 or (b) k, and m < 0;(3) ca.k, c., Ca.m, with ]c > 0, < 0 and (a) m > 0 or (b) m < 0; (4) d",c,.k, c,., with ]c > 0, < 0; (5) dt", ca.k and c,. with (a) k and > 0 or (b)]c and < 0; (6) d, d" and c,.k with (a) ] > 0 or (b) ]c < 0.From the given structure of k(V,) certain restrictions are put on these rela-

tions, as follows: the relations of type (1) are determined by the relative sizesof tl, t, t for the relations of type (2) and (3) necessarily ca.k, ca.t,for those of types (4) and (5) with tl 0 Ida, ca.k, c. , with tl > 0

a, [+ dadt ca.k, c,.t or ca.k t c,. (by Theorem 1.28)LEMMA. If in addition to the restrictions on chordal relations in k(Va) implicit

in its definition, the following relations are known: (o) all those for triples ca,ca.k, d7 and () all those for triples ca.k, ca. ca.,. such that k, l, and m have thesame sign and such that B(ca.k) B(ca.) B(ca.,.), then the remaining relationsin k(Va) are uniquely determined.

Proof. From (a) it follows that for each c,.k it is known whether k > 0 or] < 0. Further the sets B(ca.) and the element dt",. are thereby given.The relations of type (1) are implicit in the definition of k(Va).Consider a triple of type (2)(a). If B(ca.) B(ca.) B(ca.,.), then by ()

18 WILFRED KAPLAN

the relation is known. Assume then that this condition does not hold. Itfollows from Theorem 3 that, when we have determined the order relationsin V+, then all relations in V+ are known. We can therefore restrict attentionto a pair such as ca,k, ca, in V+. There are the following cases" (i) ta,k < ta,z(ii) ta,k ta, t*, with d5 B(ca,k) and d. B(ca,). In case (i) chooseso that ta,k < t0 < ta,t. Then ca, Ca.k, dt0 ]+ and ca Idat0 ca,, from Theorem 3and Theorem 1.28 respectively. Hence, by Axiom 3.2, ca, ca,k, ca,t + andca,k < ca,z. In case (ii) the same reasoning applies with t* replacing t0. Henceall relations of type (2)(a) are determined, and similarly for those of type (2)(b).

Consider a triple of type (3)(a). From the result of the preceding paragraphwe are given whether ca,k < ca, or ca, < ca,k. Suppose (without restrictinggenerality) the first case. Then (1) ca, ca,k, ca, [+. Choose to > max(ta,, ta,,). Then [c,,,, Ca,,, dto + and [Ca, Ca,Z, dto I- Hence, by Axiom 3.1,]ca, ca,, ca., +. This and (1) give, by Axiom 3.1, ca,k, ca,, ca,, +. Hencein the case (3)(a) the relation is determined, and similarly in case (3)(b).

Consider a triple of type (4). There are the following cases" (i) dt B(ca,k),B(ca,t); (ii) dt c)[. B(ca,k), B(ca,,); (iii) dt B(ca,k), B(ca,t); (iv)

dT B(c,,,k), B(ca,t). In case (i) we hve t > 0 and [ca, Ca,k, dtc,,, [dt[Ca,z, whence, by Axiom 3.2, Ca,k ]dtlc,,,,z. Cse (ii) is treated in thesame way. In case (iii)[ ca, ca,k, t, +, ca ca,z, dt 1-, whence, by Axiom 3.1,

c,,,,k, dt,a, Ca,t [+. In case (iv), if t 0, then take to > max (ta,k, ta,,). Hence[d$, Ca,k, dto [+, Ida, ca,z, dto[, whence, by Axiom 3.1, [ca,t, do, ca,k ]+.If t > 0, then c, dt lea,k, c, dtl ca,z .We know that Ca,k dt ca.z or [ca,k,dtl Ca,t But Ca,k dt, c,,,., contradicts Axiom 3.4. Hence c,,,k, dt,Thus, by Axiom 3.2, [ca,k, dt Ca,,] [Ca,k, Ca, C,,,,] [Ca,a:, d$ Ca,t]. Hence,

[+. Hence in case (4) the relations areby the case t 0 above, ca,k, d,determined.

In case (5)(a) we can assume ca,k < ca,t, whence ca, ca,k, ca,t +. This coversthe case tl 0. Suppose t > 0. There are then three cases" (i) dt(ii) dt B(ca.k) and B(ca,,); (iii) dt c.[7. B(ca,k). In case (i) we haveca, ca,t, dT [+, whence, by Axiom 3.1, c,, ca,t, dT, +. In case (ii) we have

]ca, Ca.k, dt [+ and ca dt c,,,,z, whence, by Axiom 3.2, Ca,k dt[c,,,t. In case

(iii) we have Ca dt]Ca.k, Ca dtlca,z, whence, by Axiom 3.4, ca.k dt Ca,, isd +/- by Axiom 3.2, [ca,k, dt, ca,t]not true. Hence Ca,k, t, Ca,, and,

[ca,k, ca, ca,z]. Hence [ca,, dt, ca,z [-. This covers the case (5)(a). Case(5)(b) is treated similarly.Consider a triple of type (6)(a). We can assume 0 < t < t=, the case tl 0

being covered by the information (a). We have then Ca dtalldt. Thereare (by Theorem 1.40) the following cases" (i) dtl B(Ca,k)" (ii) d B(ca,k)tl

d d +,t2 B(Ca,k) (iii) dt B(ca,k) In case (i) we have ]ca ca,k, t whence,by Axiom 3.2, ca,k dt"ldt". In case (ii) ca dt]Ca,k and ca, Ca,k, dt +.ca,k dt dt contradicts Axiom 3.4. dt ca.k ld at would imply ca Ca,k dt byAxiom 3.3, and this is a contradiction, dt dtlca, would imply caand this also is a contradiction. Hence Ca,k, dt, dt , whence, by Axiom


d dt] [Ca,k, Ca, dt] and lca,k dt, dt ]- I12 case (iii),3.2, [ca.k, l,

c, dtlca., whence, by Theorem 1.17, dt"[dtlca,k. This covers case (6)(a).Case (b) is covered similarly. The lemma is now established.

THEOREM 4. Let k(Va) Ca [J Ca, [J d and k(V’a) be subsets in thenormal subdivisions of chordal systems E and E’ respectively. Let f(a) a’ bea one-to-one transformation of k(Va) on k(V’a) with f(ca) ca, f(c,,) ca,k,f(d) d. Iffurther [al, a2, a3] If(a1), f(a2), f(a3)] for each triple al a., a3

of types (a) and (), then f maps k(Va) isomorphically on

This theorem, together with Theorem 1, will be used to simplify the verifica-tion of the isomorphism between E and the curve-family to be constructedbelow.

1.3. Intuitive outline of procedure. We shall in the following sections con-struct a regular curve-family F corresponding to the above abstract normalchordal system E. However, instead of taking the family as filling the plane, weshall take it as filling the interior of the circle and in such a way that each curvejoins a pair of points on the circumference, no two curves having a commonlimit point.

If we consider a set Va in E, we see that its structure is that of a set of chordsin the circle such that no one chord separates any two others of the set. Sucha set, with the addition of certain points on the circumference, forms a simpleclosed curve G. The elements of 0(Va), regarded as curves, would then haveto lie in the interior of the region bounded by G. Their limit points will lie onthe gaps in G left by removing the chords of V,.We can make the situation simpler by assuming that the curves of O(V,,) and

the chord ca all cross just once the radius perpendicular to ca. If we let P be apoint varying along that radius and (P) and 2(P) be the angular co6rdinatesof the limit points of the curve through P, then, as P moves from the mid-pointof ca towards the circumference, i(P) and .(P) change monotonely. (P) and2(P) have discontinuities when one chord or several chords of Va must beskipped over.Our first step in the construction will be then to determine two monotone

functions and whose discontinuities correspond to the gaps determined bythe chords of Va. These functions will place the limit points of the curves of0(Va) and leave gaps in which we can place the chords of Va. The joining ofthe limit points of the curves of 0(V,) in such a way as to get curves of a regularcurve-family is then established by a limiting process.

In this way a family corresponding to },(Vx) can be set up, with a boundingdiameter as the element C1. On the other side of this diameter we constructh(V) in the same way. Then in the segments of the circle determined by thechords of V we fit in the h(Vl,). Proceeding indefinitely in this way we fillout the interior of the circle with a regular curve-family of the desired structure.(See Figure 1.)

20 WILFRED KAPLAN

1.4. Preliminary construction of curves corresponding to c (J O(V).Notation. Let 0 -<_ to <= . Y+.(t0) (or Y(t0)) will then denote the set of

all c.,k in V+. (or V) for which t,k t0 and d"o C B(c.,). Z+.(to) (or Z-(to))will denote the set of all c, in V+. (or V) for which t, t0 and d c B(c,){}

DEFiNiTiON. If Y+.(to) (J Z+.(to) (or Y(t0) (J Z(t0)) is non-void, then d"o is adiscontinuity with respect to V+. (or V). If Y+.(to) 0 and Z.+(to) 0 (or

FIG. l. REPRESENTATION OF AN ABSTRACT CHORDAL SYSTEM BY A REGULAR CURvE-FAMILY

Y-(to) 0 and Z-(to) 0), then d" is a left discontinuity. If Y+,(to) 0 andtoZ+.(to) 0 (or Y-(to) 0 and Z-(to) 0), then dt"o is a right discontinuity.If Y+,(to) 0 and Z+,(to) 0 (or Y-(to) 0 and Z-(to) O) d" is a bilateraltodiscontinuity.

Letq(t) be areal function of tinO =< . Avalue to will be termedrespectively a discontinuity, left discontinuity, right discontinuity, or bilateraldiscontinuity of (t) according as is discontinuous at to, discontinuous at to


but continuous to the right at to, discontinuous at to but continuous to the leftat to, or discontinuous on both sides of to.LEMMA. There exist real, monotone strictly increasing functions +, (t) and q-(t)

in 0 <= whose discontinuities are precisely at the values such that d7 is adiscontinuity with respect to V, and V- respectively and that is a left, right, orbilateral discontinuity of +,(t) or q-(t) according as d is a left, right, or bilateraldiscontinuity with respect to V+.. and V- respectively. Further, these functions canbe so chosen that

0.

Proof. There are at most countably many discontinuities dt., with respectto V,. Number the corresponding set of values of as t (i 1, 2,... ),where the range of i may be infinite, finite, or void. Then let +.(t) 0 for0 < < t,, 1 for t < < ;let +(t.) 1, 0, or 1/2 according as dt is aleft, right, or bilateral discontinuity with respect to V+,. Now set

t+ E 2’

o+,(t) then has the desired properties.To obtain o(t), we number the discontinuity values t,, with respect to V

as t. (i -1, -2, ) and proceed in the same way. The lemma is thusestablished.

Suppose now that o+.(t) and q(t) are chosen as in the lemma. Then choose asequence s, of points on 0 _-< < o, everywhere dense on that infinite interval,and further including all points t (i 1, +/-2,... ). Let +k](t) (n1, 2,... ) be the function equal to o+,(t) for 0, 1, 2,... and fors, s2,--., s,, and varying linearly between these values. Similarly, let-p(t) be the function equal to,p(t) for 0, 1, 2, and for s, s, s,and varying linearly between these values. It follows from a theorem ofLebesgue that lim +".(t) +,(t), lira-k".(t) q(t). Further, since +,(t)and o-(t) are monotone strictly increasing, +,](t) and -](t) are monotonestrictly increasing, and are, moreover, continuous.

Construction of curves Dr. Corresponding to each element dt wenow define a curve /57. The curves /57 will fill the region x < 1,0 =< y < (which we shall later map homeomorphically on asemicircle) For each the curve D is defined as the graph of thefunction y if(x) in --1 < x < 1, where if(0) t, f(1 2-n)+:(t), fT(-1 + 2-) -:(t) (n 1, 2,... (0 -< < ) and fT(x)varieslinearly between these values. It follows immediately that lira f(x) +,(t),lim f(x) q-(t).

See E. Borel, Legons sur les Fonctions de Variables ROelles, Paris, 1928, pp. 97-98.

22 WILFRED KAPLAN

1.5. Construction of curves of V c. The curves /"t have limit points(q-1, +(t)) and (-1, (t)) on x 1 and x -1. Since + and are mono-tone functions, these limit points do not in general fill the lines x =t=1, butleave gaps which are half-open intervals. If, for example, the function +(t)is discontinuous to the left at t, then lim +(t) .i is less than +(t)The half-open interval [_, ) then lies outside of the range of +(t). But, bythe above lemma and the definitions of 1.4, this occurs precisely when Y(t,) 0.

Z.(t.) O, then there is a gap (i., ] in the range of +.(t), whereSimilarly, if +

lim +.(t);if Y.(t,,) O, there is a gap [r/., r/.) in the range of r/. (t);t--*t+O

if Z(t) O, there is a gap (, ] in the range of ,(t).If any of the sets Y+.(t) Z+.(t.), Z.(t.) is +/- )Y.(t.) non-void (i 1, -t-2,

we then represent its elements by corresponding curves C., in the correspondinggap on x 1 or x -1 left by the range of (1, .) or (-1, 7.). For example,if Y+.(t.) O, then Y+.(t) is, by Theorem 3, a simply-ordered set of elementsc., of V+.. To each c., we then choose an open interval ., on x 1, interiorto the interval _. < y < .. Further, we carry out these choices for allin +Y.(t.) in such a way that the corresponding intervals ., re pairwise dis-joint and that, if we order the C., by the size of the y-cohrdinates of their mid-points, then C., C., is equivalent to c., c.,. This construction is possiblesince the c., form an at most countably infinite set. It can be carried outsimilarly for all + (t.), for gaps.Z. (t.) the correspondingZ.(t.), Y- Since thegaps themselves do not overlap, the resulting intervals ., will never overlapeach other, and no C., intersects a D"t. We shall further assume the C.,chosen less in length than constants e., > O. The precise value of thewill be indicated below.

1.6. Further information on the curves/ and ,,.THEOREM 5. For each fixed a the curves D fill the region --1 < x < 1,

0=<_y< .Proof. Consider the half-strip 0 =< x =< 1/2, 0 -< y < . The curves /)’

therein join the points (0, t) and (1/2, +k(t)) by straight lines. +(t) is mono-tone increasing and lim +ffl(t) , +kl(0) 0. The equation of each line

t--o0

can be written as

(1) t(1 2x) q- 2+k(t)x y.

But for fixed x in 0 =< x =< 1/2, the left side of this equation is a monotone increas-ing continuous function of which --, as --, and 0 for 0. Henceit takes on each value y in 0 =< y < just once. This shows that througheach point of this half-strip passes one and only one curve. The same argument

--n--1applies to each half-strip 1 2- =< x __< 1 2 ,0 -_< y ,whereeach+k:(t), to (0, r+(t)), where r+line on a/ joins (0, (t)), where

+bna+l(t), since r+ is a monotone increasing function of r, and similarly to


each half-strip- 1 2-n- X --1 W 2-. Hence through each point ofthe region-1 x 1,0 =< y < passes one and only one/. This com-pletes the proof.

LEMM.. Let a curve-family be given by curves C w g(, u) (a <- <= b;c <-_ u <- d) in the (, )-plane, where g(, u) is for each u continuous in , andg(a, u) u. Let fill the closed region F" a <- <- b, g(, c) <__ <- g(, d).Then there is a homeomorphism defined on F, leaving invariant, taking the Conto parallel lines constant.

Proof. The equation g(, u) determines u as a single-valued continuousfunction of and w in F. For through each point (’0, 0) passes a unique C.Further, given any e > 0, the curves C with u uo fill a strip contain-ing (i*0, 0) in its interior. Hence, for i sufficiently small and > 0, i" i% < it,] 01 < t implies u u01 < e. Thus u is a single-valued continuousfunction h(, ).We then make the transformation on F" ’ , ’ h(, ). This trans-

formation is one-to-one und continuous, hence a homeomorphism, and takesF onto F’- a _-< ’ -< b, c _-< ’ _-< d; each curve C becomes tie curve C w u.The lemma thus follows.

THEOaE 6. To each closed interval I lying on or on a ,. corresponds apoint set G lying in -1 x 1, 0 y and such that the set G (J I Hcan be mapped homeomorphically on a region a <- <- b, c <= <= d so that theinverse image of each line constant, a, is an arc of a curve of ais I. Further, if I is onD and (xo yo) is any point such that (Xo O) is interiorto I and yo O, then G can be chosen to include (xo yo) as interior point.

Proof. If l lies on / let l be given by a __< x _-< b,y 0. Let (x0,yo)D, be the curve through (x0, y0)be given witha x0 bandyo > 0. Let -"

in virtue of Theorem 5. Take t > to and let G be the region a =< x __< b, 0 < y __<f (x). Through each point of H G (J I then passes one and only one/’The conditions of the above lemma hold and the desired homeomorphism isobtained.

If I lies on u ,,, then suppose, for example, that ,, is in the interval_

< y < , on x 1 and that I is the subinterval y0 =< y =< Yi. For N suffi-ciently large we then have s t and b,(t,) ,. Let m be the largestinteger less than t, and, if any of the numbers s, s., s_ fall in the in-terval (m, t), let s denote the largest one. Otherwise let s, m. Letfor j 2, 3, s be the s. of smallest index n- > N to fall in the interval(s._, t). We have then s < s,+ and, since the s. are everywhere dense,lims. s t,. Further, forn Nweset

h max (max (+b+(t) +b(t)), O)

for restricted so that =< +b’,+(t) =< . For n -[- 1 not equal to an n- wemust have h, 0, since +b+(t) +b’,(t) in the interval s.;, -< _-< t, where

24 WILFRED KAPLAN

n. is the largest n. < n -F 1. This follows from the very definition of +(t)and the fact that

+:(so) +"(S,o < ""If n + 1 n.+l, we have-+---n-l-i0 < h, < W, ,s,i+,) "(s,i) a( hi+ 1) i).

It follows that

h Z [(.,+,) ()](1)

< lim (.,)1

Finally, if we refer to the curves we see that is an upper boundfor the slope of the curves D in the rectangle

--n--11--2-nx1--2 ,.y:.

Now let g(x) be the function equal to (h, + 2-").for x 1 2- andn=N+l

varying linearly between these values. If we take 9(1) as lira 9(x) (whichxl--0

exists, by (1)), then g(x) becomes monotone increasing and continuous in theclosed interval 1 2-u- x 1.

Next, for any point (1, ) of the interval I let 9(x) 9(x) 9(1) + . Hencelira g(x) . Then for n sufficiently large

x 1 and (1, ) on I. The curve y g(x) then meets each curve D7 at mostonce. For the slope of y g() is always greater than that of a D at anypoint of the rectangle.

If we now apply the above lemm to the curves y 9(x) in the regionwhich they fill for (1, ) in I and 1 2 x 1, then a homeomorphism Tleaving x invariant maps them on the lines y constant, for y in I. UnderT each part of curve 7 in F becomes a curve meeting each line y constantat most once, hence a curve of the form x O(y). In particular, the curvex O*(y) through x 1 2-", y y is defined for all y in I. Through eachpoint of the region F1 0*(y) x 1, y0 y y then passes a curve zO(y) or the curve x 1 (i.e., the interval I). A second application of the lemmatherefore gives a homeomorphism T under which the curves x O(y) and Ibecome the lines x constant. Set =-z, w y, and the theorem isestablished.

1.7. Construction of the family F. Thus fr we have indicated the construc-tion of a set of curves for ech >,(V,) of E. This set consists of a set of curvesI5"t filling the half-strip-1 < x < 1, 0 =< y < and a set of intervals =.on the boundaries x 1 and x -1. We shall now map each such half-strip


on a certain region of a circle. By doing this successively for (V1), all (VI,.),all (Vl,,z), we shall obtain a family of curves filling a semicircle.For (V1) we perform the mapping in two stages, as follows. We first per-

form the homeomorphism TI on the half-strip -1 =< x -< 1, 0 -< y < . This maps the half-strip on the semicircle0 <= y’ <-_ (1 x’) minus the point x’ O,y’ 1. The line-1 x =< 1,

y’ 0. The linesx 1 andy 0 becomes the diameter -1 < x’ < 1,x -l become the arcs 0 =< , < ,1 r’ land1/2 < ’ =< ,r’-- linthecorresponding polar coSrdinates. Hence the curves/ have as images curves

joining points of these two arcs. The curves C1. have as images opensubarcs C1. of the two arcs. Let C.k denote the corresponding chords, minusend-points.We now perform a further homeomorphism. The chords C.k, plus the

points of the semicircle not on the , form an arc r’ g(t’) joining the twoends of the diameter. Our homeomorphism is now taken as T2" 0 ’,r r gl(O’). This maps the semicircle plus interior on the set 0 < r < gi(0),0 =< _-< r. The arcs , become the chords C,, the diameter

joiningremains fixed, becomes the curve C1, the curves/ become curves Dtpoints of the circle r 1.We now represent each },(VI.) in the same way as a family of curves in the

segment bounded by C,, the elements c,.z becoming chords C.. in the seg-ment. ,(V:.) is thus represented by curves filling a region g(O) <= r -< gl,(0),where 0 is restricted to the interval determined by C1, and r < 1.

Proceeding in this way, we fit curves corresponding to each h(V,) in the semi-circle, the c, being represented by chords C,. A similar process is carried outfor the h(V,*) in the lower semicircle. In all cases the homeomorphism of thehalf-strip -1 _-< x __< 1, 0 =< y is analogous to that for ,(V) above, exceptthat in all cases it must be chosen to preserve orientation.We thus obtain a family F of curves C in one-to-one correspondence with

the elements c of E (see Theorem 1.39). It remains to be shown that F actuallyfills the interior of the circle, that F is regular, and that F is isomorphic to E.

1.8. Proof that F is a regular family filling the interior of the circle. We firstremark that the numbers e,. and e*,. (see 1.5 above) have not yet been fixed.

THEOREM 7. For proper choice of the . and e*,., F fills the interior of thecircle.

Proof. Denote by G,. the region g,(O) =< r -< g,.() (r < 1, on the arcdetermined by C,.) in which are the curves of F corresponding to k(V,.) andby G the region 0 =< r =< g(O) in which lie the curves corresponding to ),(V).Similarly denote by G*,. the region g*,(O) r <= g*,.:(O) corresponding to ),(Y:)and by G* the region 0 _-< r _<. gl () corresponding to k(V*). The curves of Factually fill the regions G, and G,*, as follows from Theorem 5.,We now choose the ,. and ,. so small that the chords C,. and C*.k are so

26 WILFRED KAPLAN

small that g,(0) > 1 2-m, g,(0) > 1 2-n, where m, n respectively are thenumbers of elements in the two sequences a. We must now show that the set

’ G, (J e*, coincides with the interior of the circle.Suppose the G, did not fill the interior of the upper semicircle. Let P be a

point of that interior and not in ’ G,. Since P is not in GI, it lies in somesegment bounded by a chord Cl,k. Repeating this reasoning we find that thereis a sequence kn (n 1, 2, ...) such that P is in the smaller segment boundedby Cl.k.k...., for every n. Since g,(O) > 1 2-’, this implies that the dis-tance of P from the origin is greater than 1 2 for every n. This is impos-sible if P is interior to the semicircle. Hence the set G, covers the interiorof the upper semicircle, and similarly the set G*, covers the interior of thelower semicircle.

THEOREM 8. F is regular.

Proof. Consider an interior point P of a G,. From Theorem 6 we concludethat there is an r-neighborhood of P (one of whose sides is an interval I on

If P is on a C,,, then choose I, an interval of C,,, to contain P. There isthen, by Theorem 6, an r-neighborhood U1 in G, one of whose sides is I and oneUs in G,,k, one of whose sides is I. From the construction of F, 1 andhave only I in common. Hence (J . is an r-neighborhood.The cases when P is on C or on the lower semicircle are handled similarly.

Hence F is regular.

1.9. The chordal relations in F. The family F has been constructed as a setof curves filling the interior of a circle. Each curve C has exactly two limitpoints, which are distinct and lie on the circumference. No two curves have acommon limit point. If we regard the interior of the circle as homeomorphicimage of the plane, then we can introduce the chordal relations in F just as inthe case of a family filling the plane. Let T be a fixed o-homeomorphism of theinterior of the circle on the plane. For each triple C1, C., Ca of F we thenassign the chordal relations of T(C), T(C.), T(Ca).THEOREM 9. If C is a curve of F with limit points P and Q (i 1, 2, 3),

then CI CI Ca is equivalent to the condition that the limit points, if properlynamed, lie in the order Pa P1, P. Q QI Qa on the circumference. C1, C. Ca +

is equivalent to the condition that the limit points can be so named that they lie inthe order P, Q, P, Q, Pa, Qa on the circle, and that this order determines apositive orientation of the circle.

Proof. Suppose Ci CI Ca. Then T(C) C (T(C1)), T(Ca) *(T(C))for proper choices. Let A(C), A*(C) be the sets T-()(T(C1))), T-(*(T(CI))).Then C. A(C), Ca A*(C1). If we name P and Q in a fixed way, then C.must lie on one arc PQ, Ca on the other. Hence the other limit points can beso named that the final order is P, P, Q., Q1, Qa, Pa, as desired. Conversely,


if such a naming is possible, then C2 C A(C1), Ca C A*(C1), whenceT(C.) T(C1) T(ea) and C C1Suppose IC1, C, Ca +. There is then a positively oriented closed curve

T(M1)T(M)T(Ma)T(M1) through the points T(MI) on T(C), T(M2) on T(C2),T(Ma) on T(Ca) and not meeting T(C), T(C2), T(C3) otherwise. (See I, 2.5.)The inverse image of this curve is a positively oriented curve MM2MaM meet-ing C1, C, Ca only at M1, M, Ma. The existence of such a curve impliesthat C1, C, Ca determine circular arcs PQ, P.,.Q, PaQa which do not overlapand which, in the order given, follow the positive orientation of the circle. Thelimit points can thus be named as stated in the theorem. Conversely, if theycan be so named, then C1 C Ca, Ca C C2, C C1 Ca are all impossibleby the first part of the theorem. Hence C, C, Ca +/-. But C, C:, Caimplies by the previous reasoning that C1, C:, Ca determine circular arcsPQI, PQ:, PaQa following the negative orientation of the circle. This contra-dicts the given naming of the limit points. IIence Ct, C2, Ca + and thetheorem is established.

TIEOEM 10. The sets W. C, [.J _, C, and W*. C*. (.J _. C*determine a normal subdivision of F.

Proof. Consider first the curves of the upper semicircle. These form a setF0 C (J ti(C), by the definition of ti(C) (I, 3.2) and the preceding theorem.The curves C and C. of W are chords of the circle and are the images of theline y 0 and of intervals on x 1, x -1 of the half-strip -1 _<- x _-< 1, 0 _-<y < . It follows that no one of these chords separates any two others.Hence, by the preceding theorem, C, C, C3 +/- for an); three curves of W.Thus each W is cyclic.

For each C., ti(C,) denotes the curves of one of the two segments boundedby C.. We shall fix the choice by requiring that it(C,) does not include C.We then have

0(w.) IIwhere this equation serves as definition of 0(W.) in case W. contains only C..It immediately tollows that h(W.) and (W.,) are adjacent.Each set ,(W.) is homeomorphic image of the curves D7 and C., in -1

x -< 1, 0 _-< y < . It follows from Theorem 1.29 that DtlDtlDt isequivalent tot1 t <: taorta t tl. Hence theD"t in0 _-< arehalf-parallel.

Since each O(W.) fills the interior of G., we must have F0 h(W.).Conditions (1)-(5) of seminormlity are thus verified. To verify condition

(6) we tke any curve C., of W. C.. By our construction of the curves/t and .,, the interval (., lies below the limit point of/"t.,+l on x 1 orx -1 (according to the case). Hence no one of the three curves C., C.,,

t.,+l can separate the other two. Hence, by Theorem 9,

28 WILFRED KAPLAN

Condition (6) thus holds. Hence F0 is seminormal and similarly F0*C1 U 6"(C1) is seminormal. Thus F is normal.

THEOREM 11. E is isomorphic to F.

Proof. The elements C of F have been constructed to be in one-to-one cor-respondence with those of E. It remains to verify that this correspondencepreserves the chordal relations. By Theorem 1, we need only show that therelations are preserved in each },(V.). By Theorem 4, we need only show thatin each h(V.) and corresponding },(W.) the relations are the same for correspond-ing triples of types (a) or (/).

C(msider a triple of type (a). Suppose, for example, c., V+.. Theinterval C., then lies on x 1. Further, a limit point of curve D on x 1lies bove ., according s d B(c.,), i.e., ]c., c.,, d +. But the factthat/) has limit point on x 1 bove ., implies that the limit points of thecurves C., C.,, D, in that order, ollow the positive orientation on the circle.Hence, by Theorem 9, C., C.,, D +. If, on the other hand, the limit pointof/)$ on x 1 lies below ,, then d7 : B(c.,), and hence c. dlc.,. Theimage D" of Dt on the circle will then separate C. and C.,, whence

C.,. Thus for triples of type (a) the relations are the same.For a triple of type ()" c.,, c.,, c.,. assume, for example, that /c > 0,> 0, m > 0 and that all are in the same set Y+.(to). They are then represented

on x 1 by intervals .,, .,, ., in the same order as the c.,, c.,, c.,that is, if c., < c., < c.,, then C., lies below C.,, which lies below C.,..By Theorem 3, c.,, c.,, c., ]*. By the construction of F, the chords C.,,C.,, C., follow the positive orientation of the circle, whenceC.,,n [+. Thus, in this case, the relations of type () re invariant, and the samereasoning holds for the other cases.We thus conclude that each h(V.) is isomorphic to the corresponding h(W.)

nd similarly that each },(V*.) is isomorphic to the corresponding h(W.*). HenceE is isomorphic to F.We now finally obtain

THEOnE 12. Corresponding to every normal chordal system E there exists aregular curve-family F filling the plane with CS(F) isomorphic to E.

The family F here is the image of the bove family F under the homeomor-phism T of the interior of the circle on the plane.

2. O-equivalence of isomorphic curve-families

2.1. The equivalence and o-equivalence classes of curve-families. Let F1 ndF be two regular curve-families filling the plane. F is termed equivalent toF if there is homeomorphism of the plane onto itself transforming each curveof F onto a curve of F. This equivalence is reflexive, symmetric and transi-tive. Hence the curve-families are grouped in equivalence classes.We now make a finer subdivision. F is termed o-equivalent to F if there is

an o-homeomorphism of the plane onto itself transforming each curve of F


onto a curve of F2. The o-equivalence also divides the set of families intoclasses, the o-equivalence classes. Each equivalence class is then the union oftwo (possibly coinciding) o-equivalence classes.

THOE 13. If F1 is o-equlvalent to F2 then CS(FI) is isomorphic to CS(F=).For the chordal relations, as defined in I, 2.4 and 2.5, are invariant under

an o-homeomorphism.In the present section we shall establish the converse of Theorem 13" if

CS(F) is isomorphic to CS(F=), then F is o-equivalent to F=. From Theorem1.38 and Theorem 12 it will then follow that the o-equivalence classes are inone-to-one correspondence with the isomorphism classes of normal chordalsystems.

2.2. Map of O(V) onto parallel lines. We suppose given two regular curve-families F1 and F2 filling the plane, and further that E CS(F) is isomorphicto E CS(F2). Let f(C1) C be a fixed isomorphism of E onto E.By Theorem 1.38, E is normal. Suppose then E divided into two subsetsE C U (C) and E* C U *(C) and that E is seminormally sub-divided by cyclic subsets V for a in A and E* by cyclic subsets V* for a

in A*.Ie then immediately follows that f(E) E0 C U (C) is seminormally

subdivided by the subsets f(V) V, since seminormality is defined whollyin terms of the chordal relations, and f preserves these relations. Similarlyf(Eo*) E* C U *(C) is seminormally subdivided by the sets VFf(V*).

For simplicity, we shall from this point on use only those indices which arenecessary to distinguish the elements or sets involved. Thus V will stand forany fixed one of the sets V, V*, g, Y. A pair V, Y will mean a pairin which all other indices coincide.

Consider then a fixed (V) C [J C.k [J D. Omitting the indexk

a, we write this as h(V) Co [J C [J Dt (with Do Co). We assume thek

C numbered as in Part 1, so that/c > 0 implies C c V+, k < 0 implies C V-.THEOREM 14. There is an extended cross-section F from a point Po of Co to

in O(V), crossing all Dt of h(V) and no other curves of F.

Proof. If V has more than one element, then choose (C0) to include O(V)and each (C) to include 0(V). Then 0(V) (C0).II (Ck). Otherwise0(V) (C0) tor one choice of ,(C0). Any curve C of (C0) which can bejoined to the point P0 Co by a curve not meeting V Co is in 0(V). This istrivial if V- Co is void. If there isaCin V- Co, then neitherCIC01Cnor Co lCIC is true. Hence, by Theorem 1.28, C 0(V).By Theorem 1.32 the curves C, if infinite in number, tend uniformly to

infinity. Hence a small cross-section from P0 to point Q of (C) will meetno Ck and thus Q lies on a curve Dt of 0(V), t > 0. Let then r be the least

30 WILFRED KAPLAN

upper bound of all values such that there is a cross-section PoQ with Qon Dr.Suppose r < . Then take Q on C. A sufficiently small neighborhood

of Q will be in (C0) and in all (Ck), hence in 0(V). Thus we can drawcross-section Q2Q,.Q3 in t(V), with Q on Dt. Q. on Dt and, for proper number-ing, t. < r < t3. (See Theorem 1.29.) Further, if t. < < r, then Dr2 Dt D,whence D crosses QQQ. By definition of r, there is a t4 with tsuch that there is a cross-section PoQ4, with Q on Dr4 There is then, by thecorollary to Theorem 1.30, a cross-section Q4Qr. Since Co Dt4 Dt, PoQQQis a cross-section. This contradicts the assumption that r is the least upperbound. Hence r .We can thus choose a sequence 0 ,< t < t < t <... < t <... with

lira t such that there is a cross-section PoQ,, with Q on Dt. By the

corollary to Theorem 1.30, there are then cross-sections PoQ, QQ,...,Q,Q+, These together form n extended cross-section F PoQQQ,Q,+ in 0(V) [J Co. By Theorem 1.29 F must cross 11 D in 0 _-<hence meets only those curves.F must tend to infinity. For if F has a limit point, it cannot be on a Dr,

must hence be on a C. By condition (6) of normality, Co C, Dtgreater than some fixed t’. But Co Dr Dt+ for s > 0, whence Cand hence F could have no limit point on C. Hence F tends to infinity.

COROLLARY 1.is V.

O(V) forms an open simply-connected point set whose boundary

Proof. By the above theorem and Theorem 1.37, O(V) is an open simply-connected set whose boundary is cyclic. Since O(V) (C0).I-I (C), theboundary of 0(V) must consist precisely of the curves of V.

COROLLARY 2. Theorem 1.41 holds for the h(V) under consideration here.

For the proof, as given in I, depends only on the facts of Corollary 1 and onthe fact that the set h(V) is obtained from a normal subdivision.Remark. It can be further shown that F tends properly to infinity and

hence, from Corollary 1, that any normal subdivision of a curve-family is ob-tained by the method of I.

TEOnEM 15. There exists an o-homeomorphism T mapping Co (J O(V) onthe half-strip -1 x 1, 0 <- y so that each curve D is transformed ontothe line y t.

Proof. ChooseFasinTheorem 14. Fisacurvex x(t), y y(t) in O =<. We can extend F to a larger extended cross-section F’, on which theparameter rns from -1 to . The theorem is then proved in the same wayas Theorem 1.30. We first obtain a map onto the parallel lines filling thehalf-plane y >= 0, and an elementary transformution reduces this to the half-strip desired.We remark that it is possible by this theorem to map each O(V) on the cot-


responding 0(V2) o-homeomorphically so that the curves D become the curvesDt However, this homeomorphism will not in general transform the boundarycurves C onto the corresponding boundary curves C. In the followingparagraphs we shall ensure that the homeomorphism can be so chosen as to beextensible to the boundary.

2.3. Isolation of the curves Ck. We denote here by tk the value of previouslydenoted by t,..

THEOREM 16. A correspondence can be set up between the curves C and curvesH, with the following properties" (a) to each Ck of V corresponds one Hk (b)each H is an open curve, tending to infinity in both directions, and lying whollyin 0(V); (c) H.H, 0; (d) H.Dt 0; (e) Hk.Dt+I 0; (f) C H ]C(g) C]HIHk, (h) CklH]Dt; (i) CklHIDt+I; (j) C]HlC0; (k)[C0, Hk, D,,+] [Co, Ck, Dtk+l]; (1) [H, Hk,, H,,] [C, C,,, C,,]; (m)[Co, H, Hk,] [Co, C,, C,]; (n) [D**+, H, H,]-- [Co, C, C,]. Through-out k, k’, and k" are assumed distinct.

(Remark. The chordal relations have meaning in (f) (n) by virtue of(b) (e) and Theorem 1.25 as applied to general families of non-intersectingcurves.)

Proof. For convenience we renumber the Ck temporarily as a simple sequenceC, (p’ 1, 2,...). We define H thus" By applying a suitable o-homeo-morpbism of the plane onto itself we can assume that C1 is the line y 0,

< x < ,andthat O(V) lies in (C) taken as y > 0. The curvesCtend uniformly to infinity. It follows that the distance of the point (x, 0) onC from the set

is positive, equal to a number r(x) > 0, where r(x) is a continuous functionof x. H is.then chosen as the curve y 1/2r(x). The curve H1 in the originalplane is then obtained by applying the inverse of the homeomorphism.

Suppose we have defined the curves H for p 1, 2, n. Then, as above,assume C+1 is the line y 0, < x < , and that O(V) lies in y > 0. Thentake r(x) as the distance of (x, 0) from the set

(V C+I) U H U D+ U D++Iand H+ as the curve y 1/2r(x). Apply the inverse transformation to obtainthe curve in the original plane.The set of all H (p 1, 2, ...) will then be completely defined by induction.Properties (a) and (b) then hold, and also (c), (d) and (e), since these condi-

tions are invariant under a homeomorphism. Further, by the construction,H separates C from each of Ht H. H,_ Dt, Dt+l, Co and from allC, with p’ # p. Thus (f), (h), (i), (j) hold and (*) C H ]H,, for p" < p.If p’ > p, then by (f) C, [H C and by (*) C, [H, H. Hence, byTheorem 1.17, C,IH, H,,, Thus (g) holds.

ILFRED KAPLAN

We next verify (k). Hk c0 D+ is impossible, by (b). If Co D+ H,then (i) and Axiom 3.3 give Co[Dt+IlCk. This is impossible since Dtk+l C

B(C). If Co[Hk[Dtk+l, then this, with (i) and (j), contradicts Axiom 3.4.Hence Co, Hk, Dt+l +. This and (i) give by Axiom 3.2 [C0, H, Dt+][C, H, D+].We now verify (1). IfH H, H,,, for example, then by (g) C, ]H, [H,,

and C, H, ]H. These three relations contradict Axiom 3.4 and similarlythere is a contradiction if H, ]H [H,, or H ]H,, H,. Thus [H, H,,H,, [. This and C]H[H, give by Axiom 3.2 (1) [H, H,, H,,][C, H,, H,,], with C, H,, H,, . Similarly C, H, H:,, gives (2)[C, H,, H,,] [C, C,, H,,], with C, C,, H,, ] nd againC,, ]H,, [C, gives (3) [C, C,, H,,] [C, C,, C,,], with ]C, C,,C,, ]. Combining (1), (2), and (3)2 we obtain [H, H,, H,,] [C, C,,C,,], and (1) is established. (m) and (n) are established in the same way, bymeans of (j) and (i) instead of (g). The proof of the theorem is thus complete.The curves H enable us to isolate neighborhoods of the curve C. How-

ever, in order to ensure that these neighborhoods have a simple structure, weimprove the choice of the H by means of the following theorems. We shallassume the H chosen fixed.

DEFINITION. A curve will be said to isolate the curve C of V if it has thefollowing properties"

(Isl) is an open curve tending to infinity in both directions;(Is2) lies wholly in O(V);(Is3) .H 0 and CH(Is4) if (,) D C, and (C,) D ,, then the closed region , C, U , U

(,).(C) can be mapped o-homeomorphically on the regionW" 0 x (y+ 1)-1, < y <

so that the inverse image of each line x constant > 0 in is on a curve Dof O(V), of x 0 is C.THEOnEM 17. Let the curve isolate the curve C of V. Then in the regionW of (Is4) has as image the curve x (y 1)-. The point (1, 0) has as in-verse a point P which divides into two extended cross-sections a and .intersects precisely those curves Dt of O(V) for in an interval [t0, t) as in Theorem1.41.

Proof. Since x 0 has as inverse C, x (y 1)- must have as inversethe rest of the boundary of , i.e., . Let Q be the inverse of (0, 0), PQthe inverse of the straight line segment joining (0, 0) nd (1, 0). Any suburcof PQ containing neither P nr Q is then a cross-section. If PQ, itself meta curve of F twice, then, from Theorem 1.8, the same would hold for a sub,re

not containing P or Q. This cannot arise, hence PQ is a cross-section.Thus the inverses of x (y 1)-, y 0 and of x (y 1)-, y 0 are

cxteded cross-sections and . By Theorem 1.41 and Corollary 2 toTheorem 14 PQ meets only those Dt of O(V) of an interval [t0, t). Hencethe same holds for a and .


THEOREM 18. Let Ck be a curve of V, F0 an extended cross-section joining Coto in O(V), and e a given number > O. A curve " can be found so that" (a) itisolates C (b) ’.F0 0; (c) the length of the interval [to, t) of for whichmeets D is less than .

Proof. Suppose Dt c B(C). Then draw a cross-section P’Q joining P’on Dr, in 0(V) to a point Qk on C. By Theorem 1.41, t’ < t. By Theorem14, a point P0 on Co can be joined to Pt, by a cross-section. By Theorem 1.30,the set S formed by the curves D in 0 =< < t plus C can be mapped o-home-omorphically on a strip 0 =< x -<_ 1, < y < , so that Ck becomes x 0and the curves Dt become the lines x t t.Now H lies in 0(V). By (h) of Theorem 16, C [H Dt. If Co D [H,

then Co lD, Ck, and this contradicts D,, B(C). Hence H lies in S andhas image H in the above strip. Let I’ be the image in the strip of the partof F0 in S.

Let now r(y) denote the distance of (0, y) from the set H LI F. Since F0tends to infinity, I’ can have no limit points on x 0 and the same holds forH. Hence r(y) >. 0 and is continuous. Now set

min r(y) for Y0 _>- 0,(yo)2r(0)(y -{- 1) o__o

emin r(y) for y0 < 0.(y0)

2r(0)(y + 1) o___-<o

Then let 7k denote the curve given by x (y) and 7 the inverse image of 7k.is then an open curve.Now 3’k lies wholly in t(V) and approaches the boundary of S in both direc-

tions. It cannot have limit points on any curve of V C, for then C couldbe joined to points arbitrarily near Co or a C, with k’ k without crossing H.This is impossible by (f) and (j) of Theorem 16. By (h) we see that , doesnot approach Dt,. Hence / tends to infinity in both directions.No arc can be drawn connecting points on C and H and not crossing

For the image of such an arc would have to lie partly in the set 0 =< x < (y).But no point of x 0 can be joined to H in the strip without crossing k.Hence the image arc would have to tend to the boundary of 0 =< x -_< (y).This implies, as in the preceding paragraph, that the inverse image tends toinfinity in the plane, and we have reached a contradiction. Hence C [7 H.

If now (/) C and (C) k, then the region o C,(J(/k).(C) has as image the set 0 _-< x _-< (y), - < y < . Forthe inverse image of the region 0 =< x _-< (y) is a region whose boundary con-sists precisely of 3"k and C, which hence must coincide with w.An elementary transformation now transforms the region 0 _-< x =< (y)

onto the region W’0 =< x __< (y2 -4- 1)-1 so that each line x constant becomesa line x constant and the curve x q(y) becomes the curve x (y -4- 1)-.

Conditions (Is1), (Is2), (Is3), and (Is4) are now satisfied, hence isolates

34 WILFRED KAPLAN

C. Further , does not meet F0. Since (y) < e, 7 meets only lines xfor It t e, hence 7 meets only curves D for It t e.

In case Dt ( B(C), we proceed in the sme way, with Dt+l playing the rbleof Co and condition (i) of Theorem 16 replacing (h). Thus Theorem 18 isestablished.We now apply the theorem, remarking that the inverse under T (of Theorem

15) oflinex ,,0 <= y ,where,isaconstantwith I’1 1, isacurve F0 of the type described. We then choose for each C in V n isolatingcurve 7 not meeting the inverse of x . The precise choice of the constants

will be indicated below.

2.4. Properties of the 7

THEOREM 19. The curves 7 satisfy all the conditions of Theorem 16, if through-out Hk is replaced by 7.

Proof. Since Ck 17 H, if we choose (H) to include Ck, then 71 c (H),while by Theorem 16, (f) (j), C,, Hk,, Dtk, Dtk+l, Co all lie in *(H).Hence 71 fails to intersect any of these curves. Applying Theorem 17, weobtain (1) C 17 C,, (2) C 17 H,, (3) C ]7 Dtk, (4) C(5) C 17 C0. From (2) and (f) we conclude 7k Hk, C,. This and H,7, C, give that 7k and 7, do not intersect and further C I 17’. It isnow seen that (a) through (j) all hold with H replaced by 710. Since (k) (n)are derived from (a) (j), they must also hold with H replaced by 7. ThusTheorem 19 holds.We shall refer to Theorem 19 (a), (b), as meaning Theorem 16 (a), (b),

with H replaced by the isolating curve k.

THEOREM 20. If the subcurves and tt are properly named, then the image

of 7 under the homeomorphism T of Theorem 15 is a curve x g(y) defined inthe interval [t0, t), whereby g(y) is a two-valued function (sk(y), m(y)) with( (- x s(y), -> x m:(y) and

--1 < s(to) m(to) < 1,

--1 < s(y) m(y) l for y in (tok, t),

lim s(y) lim m(y) //1/c I,Y’*t

< < 1,

and s(y) and m(y) are continuous in the interval [t0, t).

Proof. Since 7 O(V) and by Theorem 17, it follows that zk nd haves images curves x s(y) and x m(y) defined nd continuous in the interval[t0, t) and meeting only for y t0. Hence, if z and re properly named,


sk(y) < mk(y) for y t0. Also lim g(y) +/- 1, since /k tends to infinity in bothy-*

directions. Since ,k does not meet the inverse of x , we must have limY-*

s(y) lim m(y) 1 and < Ig (y) < 1.Yt

Suppose k > 0, whence C V+, ]Co, C, Dt+ + and, by Theorem 19(k), ]Co, , Dt+ +. If lira g(y) -1, then a triangle MoMkMt+ through

Ytk

points M0" x , y 0, M" the point with minimum y-coSrdinate onx g(y), and Mt+" x , y t + 1 has negative orientation. Since Tis an o-homeomorphism, this implies Co, , Dt+ -, and a contradiction isreached. Hence lira g(y) 1 k/[ k. Similarly, if < 0, lira g(y)

Ytk Ytk

--1 k/ ]. The theorem is thus established.

TEOnEM 21. () C O(V). (b) ., 0for k ’. (c) Theimage of w C under T is the set s(y) x m(y) for y in [to t).

Proof. Since C0 C, if (C0) C, then (C0) () *(C).Hence w (C0). Similarly (C,) for k’ k. Also, by definition,

C (C). Hence C O(V), and this gives (a). (b) follows fromTheorem 19 (f) and (g) by a similar reasoning.From (Is4) it follows that the image of w: C under T is a set bounded by

x g(y) and including the line segments y constant joining x s(y) andx m(y). Hence T( C) is the region s(y) x m(y), y in [t0, t).This gives (c).

2.5. Adjustment of the transformatiens T, T’-’ The transformations T1, Tof Theorem 15 (see the convention on superscripts in the fourth paragraph of2.2) are not in general such that the two curves x g(y) and x gk(y) coincidefor each k. In the present section we shall indicate how the choices of boththe and - and 7_’1 and T can be fixed in such a way that x gk(y) and xg(y) coincide.We consider first the case k > 0. We then specify the values of the constants

,:, which will be the same for F and F:. If C1 is deiined, we take Pi SO that1/2 < 1 < 1, and then , and , under the restrictions first that -yl does not meetthe inverse under T of the line x 1 and that , does not meet the inverseunder T of x t, secondly that the corresponding intervals [t, tl) and [t0,t) are the same. This latter is possible by virtue of the fact that tl tby the isomorphism of F1 and F., that tl < tl or tl > tl according as t0 < t or

tl > tl respectively by the isomorphism and Corollary 2 to Theorem 14, andfinally that t011 and ll can be taken arbitrarily close to tl by Theorem 18. Thuswe can write ttNext (if C is defined) we choose a constant , -] < . < 1, and further

larger than both gl(t0i) and g(t0i). Then choose / and / so that they do notmeet the respective inverses of xt0, t t t., and also so that t0 t0.

3 WILFRED KAPLAN

In general, if we have defined ’1, /p, then we choose ’v+l so that 12-p- < p+ < 1 and so that + > g(t0), +1 > g(to). Then choose’v+ and "r+l not to meet the inverses of x v+, 0 =< y o and so that

while to,+ tok for k 1, p.t01,p+l t02,pq-1 t0,p-I-1, tpTi tVq-1 tpq-1,With these choices fixed, we then see that x gk(y) lies to the right of x rk,

while the line x vk+ lies to the right of the point P" y t0k, xFurther lim , 1, vk+l > ,k, and to t0k, for k k’. These relations hold

for all k 1, 2,For k < 0 we proceed in a similar way, choosing the constants _, v_z,

and the curves -1, ’-, ’-, ’-, so that -1 q- (1/2) > rk > -1,and that x gk(y) lies to the left of xthat vk_ < g(tok) and

_< g(tok),

while lim rk -1, rk_l < k, t t tok, t t but t0k t0k, for k k’.

ThEOrEM 22. If k’ k" k’/[ ]’ k"/[ k" I, and y is such that bothand g,,(y) are defined, then either

Sk, (y) <--_ mk, (y) < s, (y) <-- mk, (y)

[s, (y) <= mk, (y) < sk,,(y) <= m,,(y)

or else

__< mi,,(u) < =<__< < __<

Proof. Case I. / > 0. We can suppose (without restriction) that k’ < ]".

Suppose first that tk, t,,. The two intervals [to,, tk,) and [t0k,,, tk,,) mustoverlap. If t0k, is contained in [tok,,, tk,,), then, by the above construction,g,,(to,) > gk,(0k,). By continuity this must hold for all y in the commoninterval, since the curves do not intersect. If tk, is contained in [t0,,, tk,,), then,as y t, g,(y) -- 1, while g,,(y) ---) gk,,(tk,) < 1. Hence near t, g,(y)q,,(y) and from continuity this holds throughout. Next, if one but not bothof t0k,,, t,, is contained in [t0,, tk,), then either t, or t0k, is in [t0k,,, tk,,). Thisreturns us to the previous case. If both t0,, and t,, are in [t0,, tk,), then neart,, g,,(y) > g,(y) and by continuity this holds throughout. Thus in all thesecases the order of the points is determined purely by the intervals involved andthe fact that/’ < k". It follows that the points x g,(y) and x g,,(y)lie in the same order, for any y in the common domain of definition of g,(y)and g,,(y) (which is the same as the common domain of g,(y) and g,,(y)).Next suppose tk, t,,. The only possible cases are t0k,

or else t0k, :> tk,, t0,, > tk,,. Consider the first of these cases. Supposeto, < t0k,, < t,. At t0k,, we must then have either g,,(to,,)

< ,) <mk,(t0,,) or g,,(t0.,,) > m,(t0,,) => s,(tok,,). For if sk,

(t01,,), then by Theorem 21 (c) wk, contains ,, This is impossible by thefact that o,.:,, 0, by Theorem 21 (a).

If g,,(to,,) < g,(to,,), then it is possible to find closed curve through the


points (g,(to,), 0), (gk,(tok,), tok,), (g,,(to,,), to,,), meeting the curves y 0,x g,(y), and x gk,,(y) only at these three points, and lying wholly in thehalf-strip xl < 1, 0 -< y < 1

Moreover, such a curve has positive orienta-tion. Applying the inverse of and the definition of the chordal relations, weobtain that C, ’,, ’,, +. By Theorem 19 (n) [C0 ,,, ,,,] [C0 C,,C,,]. It follows that we can have gk,,(tok,,) < gk,(to,,) only if IC0 C,, C,,

Similarly we find gk,,(to,,) > gk,(tok,,) only if lC0 C,, C,, I-. It thus followsfrom the isomorphism that g,,(to,,) < g,(tok,,) according as g,,(to,,,) < g,(to,,,).By continuity the same condition holds for any y in the common interval.Thus, under the assumption t0,, < t0,, < tk,, the desired inequalities hold.Exactly the same type of discussion holds if t0k,, < t0,, < t,.A similar discussion holds in case t0, > t,, and t0,,, > t,,,. Thus in all

cases under Case I the inequalities hold.Case II, k < 0. The same reasoning holds, and thus the theorem follows.

THEOREM 23. T can be so chosen that it satisfies Theorem 15 and furtherthat g(y) coincides with g(y) for all k.

Proof. Case I, k > O. We shall first make certain extensions in the domainof definition of the curves g(y) and g,(y) so that the inequalities of Theorem22 continue to hold, with the exception of a possible equality instead of in-equality in certain cases.We make these changes in two stages. First (A) suppose g(y) has interval

of definition t01 -< y < tl. It may happen that (A1) some other curve xg(y) crosses the line y t01 between x gl(t01) and x 1. (It will thenactually cross the line, since to t01.) If so, we set 1 0 and leave g(y)unchanged. The same situation then holds for g(y), by Theorem 22, and againthere is no change. If this condition fails to hold, then there are two possibili-ties" either (A2) for i sufficiently small no curve g(y) (k > 1) meets the rec-tangle to1 <= y <= to1, g(tol) _-< x =< 1, or (A3) no such ti can be found. Inthe first of these cases we leave the curves g(y) and g(y) unchanged at thisstage and take til 0.

In case (A3) we choose til > 0 so small that the following properties hold"(a) no other curve g(y) meets the line segment x g(tol), to1 1 <= y <= to1(b) no other curve gk(y) meets the line segment x g(t01), t01 -/tl _-< y t01(c) there is a curve gl (Y) crossing the line y t0 -til in the interval g(01) <

x < 1 and no other curve g(y) crosses the line in the interval g(tol) < x <g,(Y)

(d) for the same k, g(y) has the analogous property for g(y).In order to find this il, we first restrict/tl to be so small that at least (a) and

(b) hold. This is possible since the curves g(y) and g(y) tend uniformly 4o

the boundary. Next let be the go(Y) crossing the line y t01,0 xg(t01) with maximal value of x, or else, if there is no such go(Y), set l(y) 0.(y) will be determined in the same way, and the index k0 will be the same inboth cases, by Theorem 22. We now restrict il further to be so small that

38 WILFRED KAPLAN

t01 1 =< y _-< t01 lies within the interval of definition of e(y) and further sosmall that no curve g(y) crosses the region (y) <= x <= g(tol), tol <- y <-_ toland that no other g(y) crosses the region ,(y) <= x <- g(to), to <- y <- to.Finally, since infinitely many curves g(y) meet the rectangle t0 _-< y _-< t01,g(to) _-< x =< 1, we can take t further so small that a gl(Y) crosses the liney to , g(tol) < x < 1, with t0 t0kl, and that no other g(y) crossesy t0 between x g(t01) and x gl(to). It follows from Theorem 22that, with the same 1 and kl, the function g.l (y) has the same properties rela-tive to g(y). The conditions (a), (b), (c), (d) are now satisfied. We now ex-tend the definition of g(y) by setting g(y) g(to) for t0 _-< y =< t01,and similarly g(y) g(to) in the same interval. Further we set s(y)m(y) g(y), s(y) m(y) g(y) in this interval.

In case (B) t < t01, then we have the analogous discussion for cases (B1),(82), (83).Now we extend this process by induction to all curves g(y) and g(y). Thus,

if it has been carried out for ] 1, 2, r 1, then we consider g,.(y) andgr(y). The discussion is then exactly as with g(y) and g(y), except that ineach reference to the other curves g(y) and g(y) for/c < r the curves as extendedunder (A3) or (B3) will be meant.As a result of this, we obtain a new family of curves g.(y) and g(y) whose

definition intervals are [t0k - tk, tk), with >= 0, so that no two curves intersectand the inequalities of Theorem 22 still hold. Finally, as a result of thosechanges under (A3) and (B3), for the new end-points [01 t0 =i= the onlypossible cases are (A1) and (A2), (B1) and (B2).We now carry the extensions one stage further. All references will be to the

curves as they stand after the above extensions. For g(y) in case (A1) wechoose g(y) as the curve g(y) crossing y 0 between x g(tol) and x 1with minimum value of x. gl(Y) will have the same property relative to gl(y).We next choose > 0 so small that no other g(y) crosses the region 0 ti <y =< Ol, g(o) <= x <-_ g(y)and no other g(y) crosses the region 0- _-< y _-<o, g(t01) <= x <__ g(y), ti being also taken so small that g(y) is defined ino ti =< y =< o We then extend g(y) and g(y) by setting

s(y) m(y) g(y) o -- Ys(y) +(1ol--ys(y) m(y) g(y)

is,(y) + ,1

o Y) g(o)-)tol y g(to)

for 0 =< y _-< 0. The extensions to the functions lie within the aboveregions and on the extensions g(0) =< g(y) < g(y) except for y 0 ,when g(y) gl (y). Similarly g(y) <: g (y) except that g(0 ) g(0). The inequalities of Theorem 22 thus continue to hold, with a new inequalityat one point, namely, at the new end-point of the interval of definition of g(y)and g(y).


In case (A2) we proceed similarly, replacing gl (Y) by the function l(y) 1.For the new functions g(y) and gl (y) we then have g(0 ) g(0 8) 1.Since no curve g(y) (for k > 1) crosses the rectangle 0 -< y -_< 0, g(0,) =<x _-< 1, and similarly no g(y) crosses the rectangle 0 -< y _-< 0, g(01) -<x -< 1, we conclude that the inequalities of Theorem 22 still hold.

In cases (B1) and (B2) we proceed in the same way.We now extend this process by induction to all curves g(y) and g(y) for

/c > 0, as above in cases (A3) and (B3), using at each stage the already ex-tended curves for the lower indices. Inasmuch as the values x g(0) g(to)tend to 1 as k -- , and on the extensions to the curves x ->_ g(0), the exten-

FIG. 2

sions (if infinite in number) tend uniformly to the boundary x 1. Further theorder relations of Theorem 22 will continue to hold. (See Figure 2.)Now as a result of these extensions the set of curves x g(y) forms a set of

curve arcs, each of which joins either a point on an arc to a point on anotheror a point on the line x 1 to a point on an arc, and there are no free end-points in x < 1. Further the intersection points of the arcs are either countablyinfinite or finite, and, if infinite, tend to x 1. At each intersection point atmost a finite number of arcs meet. It follows that these arcs divide the region0 < x < 1, 0 =< y < into an at most countably infinite number of connectedregions. The same holds for the curves x g(y).The interior of each region determined by the x g(y) has the structure

,,(y) < x < (y), where e(y) and (y) are continuous functions in the same

40 WILFRED KAPLAN

finite or infinite y-interval. That (y) is continuous and well-defined followssince on each of the curve arcs x is a continuous function of y. To determine(y) we take any point (x0, y0) of the region under consideration and let ((y0),y0) be the boundary point to the left of (x0, y0) on the line y y0 If (y0) 0,then (y) 0. If (y0) 0, then the rest of the boundary x (y) can beobtained by following along the curve arc through ((y0), y0) letting y increaseand decrease indefinitely. If a multiple point is reached, then the continuationarc is chosen to have maximum possible value of x. If, as y approaches a valueyl, (y) approaches 1, then (y) ceases to be defined beyond yl (above or below,according to the case).(y0) can be defined (in the same interval as (y)) as the function equal to 1

if there is no curve arc meeting y y0 to the right of (x0, y0) and as equal tothe minimum possible x-coSrdinate of such a curve arc when it exists. (y)is then necessarily continuous when less than 1. If (y0) 1, then (y) re-mains continuous. For otherwise the curves x g(y) would have a limitpoint on y y0, x0 < x < 1. Since the x g(y) tend uniformly to 1, this ispossible only if some x g(y) crosses the segment, contrary to assumption.

Since the functions (y) and (y) are continuous, they must form the wholeboundary of the region.Now the same discussion holds for the regions determined by the x g(y),

and, moreover, the regions are in one-to-one correspondence with those of thegk(y) in accordance with the inequalities of Theorem 22.We can now obtain an o-homeomorphism of the set 0 _-< x =< 1, 0 _-< y <

on itself by the transformation"

0forx 0, lforx 1;T"= (y) for x k(y);

2 m(g) for x m(g);

with 2 varying linearly with z between these values. This transformationleaves each line 0 invariant, is also monotone increasing and eontinuousin z for fixed g g0, henee gives a homeomorphism of each line y g0 ontoigself, and is gherefore one-to-one everywhere. Further it akes each region,() < x < (g) determined by the (g) onto the corresponding one ,() <x < (g) determined by the g(). Since the () and g2(g) are continuousfunctions, and f/ varies linearly between ,() and (g), is a homeomorphismin eaeh region plus boundary. T is thus a homeomorphism on the region0 -< z < 1, 0 =< y < , and, sinee ? , also on the boundary z 1. T isalso an o-homeomorphism.

In a similar manner in Case II, k < 0, an o-homeomorphism T of the region-1 _-< z _-< 0,0 _-< y < on itself ean be defined leaving the line z 0andeach line y y0 invariant and transforming all the x g(y) for/c < 0 on thecorresponding x g(y). The resulting transformation on the whole half-


strip --1 _-< x -< 1, 0 =< y < is then single-valued and an o-homeomorphismof the set on itself.The transformation T can now be chosen to replace the original choice

of T1, and Theorem 23 is established.

2.6. Map of F1 on F2. We assume the transformations T of Theorem 15 andthe k are now fixed so that Theorem 23 holds.

LEMMA. Let W be a homeomorphism of Co onto the line y O, -1 < x < 1such that the homeomorphism T-1W of Co on itself preserves orientation on Co.Then there exists an o-homeomorphism of Co (J O(V) on the strip -1 < x < 1,0 <-_ y < o such that T Talongeach’rk, T WalongCoandtheimageofeach curve D is the line y t.

Proof. By Theorem 20, Ikl < Igk(Y) < 1. Also I1- (1/2 Ikl) < [-Hence the distance r(x) of a point (x, 0) on y 0, -1 < x < 1 from the pointset formed by all the curves x g(y) is greater than 0. The function y (x)in-1 __< x =< ldefinedby

1 x0b(x0) min r(x), Xo >= 0;2 O<_x<_x

1 Xo(x0) rain r(x), xo <= O,2 xo<_x<_O

is continuous, monotone strictly decreasing for x >_- 0, monotone strictly in-creasing for x _<_ 0, and no curve x g(y) intersects the set 0 _-< y <- k(x),--1 _-<x=< 1.Since T-IW preserves orientation on Co, the "transposed" transformation

T(T-IW)T-1 WT- of -1 < x 1, y 0 onto itself must also preserveorientation and can be extended continuously to the end-points, which it leavesfixed. Such a transformation can be extended to the set 0 =< y =< (x), -1 _<_x __< 1 so as to leave each point of the curve y b(x) fixed and in general toleave y invariant. We write the extended homeomorphism as

TO. x (x, y), yO y.

If we define T as the identity outside of this set, then T becomes an o-homeo-morphism of -1 _-< x =< 1,0 _-< y on itself. The product - TT thencoincides with T along each ,, rl’(Dt) is the line y and along Co,TOT WT-IT W.THEOREM 24. There is an o-homeomorphism of each set o on the corresponding

D Dset such that the image of each arc of a curve in is an arc of the cureand that is of Ci of is of is

Proof. By condition (Is4) and Theorem 17, k and can both be mappedo-homeomorphically on the set W" 0 -< x _-< (y - 1)-1, - y so thateach line x constant > 0 in W is an image of an arc of a Dr, of x 0 is C.

42 WILFRED KAPLAN

Further, by construction, the intervals [tok, tk) are the same for / and .Hence by suitable choice of the o-homeomorphisms of ok and wl0 on W we can,obtain that each arc n in w has the same image x constant in W as the.arc D. If we show further that a and a, # and have the same images,hen the theorem will be established.Now is defined as the inverse of x sk(y), # of x m(y), of x s(y),

t* of x m(y) (without the extensions to these functions temporarily intro-duced in the proof of Theorem 23). From Theorem 21 it follows that the sets(y) <= x <- mk(y) can be mapped o-homeomorphically on W minus the linex 0 so that x gk(y) becomes the curve x 1/(y -b 1), and so that eachline y becomes a line x constant. The set s(y) <- x <-_ m(y) can simi-larly be mapped on W minus the line x 0. Further, ------ m(y)ink(y), by Theorem 23. Since both of these above homeomorphisms preserveorientation, we conclude that s(y) and sk(y) have the same image, i.e., eitherx 1/(y2 + 1), y _-> 0orx 1/(yZ - 1), y _-< 0, and thatm(y) andm(y)have the same image. Hence a and a have the same image, and # havethe same image. The theorem is now established.

THEOREM 25. There is an o-homeomorphism of X(V1) onto X(V) taking eachC anto Co C onto C, D onto Dt

Proof. Under T1, the set (h(V) ) [J 1 is mapped o-homeo-

morphically on the strip- 1 x 1, 0 _-< y minus the sets s(y) xmk(y). By Theorem 23, under T, (h(V) ) (J is mapped o-homeo-

morphically on the same set. In both cases the curve D has as image the liney t. It follows that (T)-TI takes (},(V) ) [3 ’ . o-homeomor-phically onto (},(V2) ) U ,. and takes D onto D. Further theimage of each -y is , and, furthermore, of a is a, of # is #.By Theorem 25, there is an o-homeomorphism of each w onto 0. More-

over, this homeomorphism coincides with (T)-T along ak, since meetseach D at most once, at point Pt, and each point pit on has image Pt onD under both transformations. Thus they coincide along , and similarlyalong , hence on ,. But is the boundary of 0 in 0(V). Hence byTheorem 21(b) the transformation (T)-T can be extended to be single-valuedover all the sets w (J C, becomes an o-homeomorphism of h(V1) onto h(V)with the desired properties.

THEOREM 26. F can be mapped o-homeomorphically on F by a transforma-tion T such that T(C) C f(C) for each curve C of F.

Proof. By Theorem 25, there is an o-homeomorphism T of X(V) onto X(V)preserving curves of the family. There is a similar transformation Tl,k Of each},(V.) onto the corresponding },(V.). },(V) and },(V,) overlap only alongthe curve C,. Hence T-,. T defines a homeomorphism of C, onto itself.Since C. is an open curve tending to infinity in both directions, and since Tl,k


and T1 both are o-homeomorphisms, T.TI must leave orientation invarianton Cl,.

,k is defined by means of an o-homeomorphism Tl.k of CI, [J O(V .) ontothe half-strip-1 < x < 1, 0 -<_ y < and an o-homeomorphism T.I ofC, [J O(V,) onto the same strip. Thus T, (,) T, on the set((v,) ,) v,. thr ,.T Wdn trortio,

of C, onto y 0, -1 < x < 1 with the property that

(T,)-W (T -T, ,T (T,)-leaves orientation invariant on C,. It follows, by the lemma proved above,that we can replace T, by a new o-homeomorphism , which has all theproperties of ,plus the property that , W along C,. If we assume T,hoe a this ,, th ong C, , ,T or T (T,)-T: ,Making these restrictions on the T,, we conclude that the transformation

T can be extended to a homeomorphism of the set (V) (V,) ontoC C(V) (V,), taking each curve onto the curve f(C). Pro-

ceeding by induction and making similar restrictions on the T, we see thatthere is an o-homeomorphism T of E onto E with T(C) C f(C). Thesame reasoning gives an o-homeomorphism T* of E* onto E* with T*(C)C f(C). E* and E have the one boundary curve C in common. Anotherapplication of the above lemma enables us to assume that T T* along C.Hence T can be extended to an o-homeomorphism of the plane onto itself,taking F onto F. Theorem 26 is thus established.We now conclude

THEOREM 27. If tWO regular curve-families filling he plane deermine iso-morphic chordal systems, then they are o-equivalent.

3. Classification of the curve-families

3.1. Abstract classification. Let be the set of all o-equivalence classes(see 2.1) of regular curve-families F filling the plane. The relation )f iso-morphism groups all normal chordal system E in disjoint isomorphism classes. Let @ be the set of all classes . On the basis of Theorem 1.38 and Theorems12, 13, 27 we conclude

THEOREM 28. There is a one-to-one correspondence w(’) between the setsand such that, for any F in and any E in w(), CS(F) is isomorphic

to E.

We have remarked in Part 2, 1 that each full equivalence class * of curve-families generates in general two different o-equivalence classes, and -’:9:* 9:U Y’. The families of one class, e.g., ’, are obtained by applying onefixed non-orientation-preserving homeomorphism T (for example, x’ x,y’ -y) to the families of the other class, ’. Such a transformation willleave invariant the relations C1 C2 C but will replace ]C1, C., C3 [+ by[Ct, C, C’ I- In special cases it may happen that the class [Yis invariantunder such a transformation, in which ease

4Ax WILFRED KAPLAN

We can now introduce a broader equivalence among the normal ch.rdal sys-tems, letting two be congruent if they are isomorphic or if one is obtained fromthe other by a one-to-one transformation which leaves the relation a[btcinvariant, but replaces a, b, c + by a, b, c I-. We then obtain a one-to-onecorrespondence between the congruence classes g* of normal ehordM systemsand the equivalence classes * of regular curve-families.

3.2. Classification by explicit chordal systems. We first point out that, as aresult of Theorem 27 and of Theorem 12 and its proof, we can assert

THEOREM 29. Let F be a regular curve-family filling the plane. There is ano-homeomorphism of the plane onto the interior of the unit circle such that thefamily F is transformed onto a regular family F1 of curves filling the interior of thecircle and with the property: each curve of F has a unique pair of limit points onthe circumference, no two curves have a common limit point.

For the family F of curves constructed in the proof of Theorem 12 has thestated property. This family could, moreover, be constructed to have a chordalsystem isomorphic to any given chordal system, in particular to that of F. ByTheorem 27 we can now map F o-homeomorphically on F.

If we now join the end-points of each curve of Ft by a chord, we obtain afamily of chords in the circle. No two of these chords can intersect, for then theend-points of one curve of F would separate the end-points of the other curveand the corresponding curves would have to intersect. By Theorem 9 we canimmediately introduce the chordal relations in the set of chords in terms of theorder of their end-points. This gives a chordal system. It is further iso-morphic to CS(F). We have therefore

THEOREM 30. To each normal chordal system corresponds an isomorphic ex-plicit chordal system.

We can thus replace the abstract normal chordal systems by the concreteexplicit normal chordal systems. It remains to determine when two such explicitchordal systems are isomorphic or congruent"

DEFINITION. Let A and A’ be two point sets on the circumference of theunit circle. Let f(P) P’ be a transformation of A on A’. f will be said to bemonotone if f is one-to-one and if for every ordered triple P1, P2, P3 of points

THEOREM 31. Let KI and K2 be two explicit chordal systems of the unit circle:x y2 1. Let A be the set of end-points of the chords ofK (i 1, 2). ThenK1 and K are isomorphic if and only if there is a monotone transformation f ofA on A such that the image of a pair of end-points of a chord of K is always apair of end-points of a chord of K.. K and K2 are congruent if and only if Kis isomorphic either to K. or to the image K ofK under the transformation x’ x,y’ --y.


This theorem follows from the expression of the chordal relations in terms oforder of end-points, as in Theorem 9.Theorem 31 can be regarded as the basis for an equivalence (or o-equivalence)

criterion for curve-families. If we imagine an associated isomorphic explicit.chordal system K for each family F, then two families F1 and F2 are equivalent(or o-equivalent) according as the corresponding systems K1 and K2 satisfy the con-gruence (or isomorphism) conditions of Theorem 31.

APPENDIX. Applications of the classification theory to the cases withsingular points

1. The fact that we have made a certain classification of regular curve-familiesfilling the plane, or, what is the same thing, the sphere with one singularity,enables us to attack the cse of families on the sphere with many singularities.We give here an illustration.

DEFINITION. An n-cyclic chordal system is a chordal system E which is iso-morphic to a subset },(V1) V [J 0(V1) of a normal chordal system E1, where Vis a cyclic subset of the system El.The class of n-cyclic chordal systems can be studied in detail by means of the

general theory of normal chordal systems.Now take as our example regular curve-family F filling the sphere with the

exception of a closed set A of singular points. Further we assume there is aclosed curve 1" on which all points of A lie and that the complementary setF A on F is composed of curves Cn (n 1, 2, of F. The set A need notbe countable. We assume it contains at least two points.To analyze the possible structure of such a family we first consider one region

R bounded by I’. (The theory for the other region is the same.)First we map F plus R homeomorphically on a circle F’ plus interior R’.

Next we join the end-points of each arc to obtain a chord K Then (asin the proof of Theorem 12) we transform F’ plus R’ homeomorphically onto theset R" bounded by the curve F" formed by the chords K’ and the set A’. Thesubset F0 of F in F [J R is thus mapped homeomorphically on a curve-family Ffilling the interior of the curve F" plus the chords K’. We extend F to afamily filling the interior of the circle simply by filling out the segments boundedby the KP by chords parallel to K’ The result is a regular family filling theinterior of the circle.We conclude that the curve-family F is n-cyclic. But F is homeomorphic

image of F0. Hence the structure of F0 is the same as that of an n-cyclicfamily, to which our previous theory can be applied.The following converse can be established, by means of recent results of

Adkisson and MacLane (see this Journal, vol. 6(1940), p. 216 ft.): given anyn-cyclic chordal system E V O(V), whose "boundary" V is isomorphic tothe set V’ of chords K’n, then there is a regular curve-family F filling R plus theC whose structure is that of E; i.e., F0 is isomorphic to E.

46 WILFRED KAPLAN

2. Generalization. Suppose now that we have a regular curve-family F fillingthe sphere except for a closed point set A which lies on a closed set F such thateach component region R of the complement of I’ on the sphere is a simply-connected region bounded by a closed curve Fi c F and that F A is formedof curves of F. To each region R [J F can then be applied the same reasoningas above. We obtain again n-cyclic families.

UNIVERSITY OF MICHIGAN.

regular curve-families filling the plane, ii

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