Regular Families of Curves I

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MATHEMATICS: H. WHITNEY

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REGULAR FAMILIES OF CURVES. I* BY HASSLER WHITNEYt DEPARTMENT OF MATHE1MATICS, PRINCETON UNIVERSITY Communicated February 10, 1932

Given a family of curves, such as the paths of particles in a steady motion of a fluid, it is often convenient to introduce a continuous function x= g(x, t), with the interpretation that the particle at the point x has moved to the point x' after a time t. It is also useful to have cross-sections of the curves; if the curve through any given point has a direction and this direction varies continuously with the point, we can construct a cross-section in the neighborhood of a point by drawing rays at that point perpendicular to the direction of the curve. In this note we associate with a general point set a function A; with its help we can, given a family of curves satisfying very weak conditions, define a function and find cross-sections as above. Examples show the usefulness of the function in topology. Conditions under which a family of curves is "regular" will be given later; the question of orienting the curves and of introducing invariant points will also be discussed. The Function ,u(S).-Let R be any totally bounded' metric space (for instance, a bounded part of Euclidean n-space), and let S be any closed' subset of R. Immerse any set of n > 2 points pi, p2, ...* p,n in S; with this set of points Pn is associated a number2 d(P") = min p(pi, pj), the s$j smallest distance between pairs of the points of Pn (which is 0 if some two points coincide). This number d(P") is certainly - b(S). Hence, for all choices of Pn, there is an upper limit for d(Pn), which we call 1A(S):

A#(S)

=

lim d(P") Pn

=

lim min p(pi,,p). PM i

j

(1)

We define Iu(S) by the equation

,A4()

=

Mn(S)2

n=2 2n1

(2)

Some obvious properties of the functions are: /lM(S) 2 uA+ (S); lim juM(S) = 0; A2(S) = b(S); b(S)/2 _< (S) 1 b(S).

n

co

Suppose S' C VE(S). Take any n _ 2 and any PM' = p"', ..., pM' in S'. Choose pi in S so that p(pi, pi') < E, i = 1, . . ., n. Then p(pi', p,') < p(Pi, Pa) + 2e, hence d(PM') < d(PM) + 2E, and yM,(S') < AMn(S) + 2e. Summing over n, we have

Au(S')

,uu(S) + 2E, if S' C V.(S).

(3)

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If S is compact also, we can write p (S') < ,u(S) + 2E. Suppose S C S', S # S'. Let p be a point of S' not in S, at a distance d from S say, and choose n so large that An(S) < d. Take i > n so that Aui_I(S) > gui(S). Take any set of i - 1 points Pi_ in S. Add the point p, forming a set of i points Pi' in S'. Now d (Pi_ 1) < ui - 1(S) < d; hence d(Pi') = d(Pi-1), and thus /i (S') _ pi-l(S) > l.i (S). As Mk(S') . Ak(S) for all k, we have, on summing, (4) ,u(S') > ,u(S) if S C S', S # S'. The last two results we can state in the forin: IA(S) is a continuous and increasing function of S. Regular Families of Curves.-We shall use the word curve in any of three senses: The topological image of an open or a closed line segment or of a circle. We shall say that a family of curves forming a totally bounded metric space is regular if the following condition is satisfied: Given any arc pq of one of the curves C and any e > 0, there is a a > 0 such that if p' C V,(p) lies on a curve C', then there is a point q' on C' such that the arc p'q' has a parameter distance (6cart) less than e from the arc pq. By this we mean that there is a homeomorphism between the two arcs such that if r and r' are any two corresponding points, then p(r, r') < e. It may be proved easily that in a regular family, no two of the curves intersect. Take any point p on a curve C of a regular family and consider the arcs pq, q on C variable. For each point q there is a ,u(pq). A(pq) increases continuously as q moves away from p; hence for a given number a not too large, there is a point q such that A (pq) = a. We write then q =

g(p, a):

(5) q = g(p, a) equivalent to ,u(pq) = a. For given p and a there are either two, one or no corresponding points q-two, one on each side of p, if p is interior to C and a is small enough. Continuity of g.-Given any point p on C and a point q interior to C, where ,u(pq) = a, and given an e > 0, take first ,B < e/2 so small that there are points q, = g(p, a - 31) and q2 = g(p, a + 3/), and the arc qlqq2 C V./2(q). Take a so small that for any point p' in Va(p), there is an arc p'q2' whose parameter distance from Pq2 is < 1. Using the correspondence thus given between pq2 and p'q2', let ql' be the point corresponding to ql. Now each point of p'ql' is within 13 of the corresponding point of pql, and thus p'ql' C Vs(pql); hence M(p'ql') _ IA(pql) + 21 = a - 13. Therefore if q' = g(p', a'), a - a' < 1, the point q' lies beyond ql'. But also pq2 C Vp(p'q2'), and hence /(p'q2') 2 M(Pq2) - 21 = a + 1; q' lies therefore before q2'. It lies consequently on the arc ql'q2'. But qj'q2' C Ve(qiq2) C VP+,12(q); hence q' C V.(q). We have now proved that given a point p, a direction on the curve C through p, and an a such that q = g(p, at) in the given direction from p is interior to C, then for every e > 0 there is a

I

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a > 0 such that if p' c V1/(p) and |a - a' < 6, then there is a point qI= g(p', a') C Ve(q). If, p and q being fixed, e is taken small enough, then there is such a point q' only in a single direction from p'. The function g is, of course, uniformly continuous over any closed set of points p and closed range of values a, by the Borel theorem. Cross-Sections.-Given p interior to C, take q and q* interior to C and on opposite sides of p, and take a so that W2f(q) and W2,(q*) do not intersect. Take any e less than any of the numbers 2a, p[W2a(q), W2,(q*) ]/2, p[p, W2a(q)], p [Pt W2a(q*) ]. If 61 is small enough, then any point p' of C' in Wa,(p) lies on a unique arc running from W2,,(q) to W2,,(q*), this arc being contained within an arc q/q*/ whose parameter distance from the arc qq* is < e, with p and p' being corresponding points on the arcs; moreover, this holds only if the arc of C' be taken in the correct orientation. We shall find a a _ '1 and a point set S with the following properties. (1) Each arc through a point p' of Wa(p) as above defined contains exactly one point of S, and every point of S lies on such an arc; also p C S. (2) S C Vj(p). (3) S is closed. The set S we say forms a crosssection through p. We shall now determine a cross-section S. Take v < e so small that if we follow C from p toward q to the first point s in W2,(q), then sq C Vg(q). Given any point p(i) near p on the curve C), let r(') be the first point of C(') starting from p(i) in Wj(q); in particular, r is that point on C. Put U = W20(q) and say ,(U 4- pr) = a. Take 1 < 1/2 so small that if pi and p2 are the points on C such that ,u(U + pir) = a - 23, ,(U + p2r) a + 2,B, then piPP2 C Ve/2(P) Take ay so small that if pi' C V71(p1), then p[g(p1, t), g(pl', t)] < (,I O < t < ,14(pq). We shall show that U -+ p11t' C VO(U + p1r); it will follow that (6) ,u(U +- pi'r') . ,u(U +- pr) + 2,B = a. It is sufficient to show that p1'r' C V,(U + p1r). If x' is any point of pl'r' and x' = g(pl', t), put x = g(pi, t), using the same t. We shall show that x C U + p1r; then p(x, x') < 13, and the inclusion follows. If ql' = g(pi', ti), where t1 = O (p1q), then p(q, ql') < 13 < 7i. Thus ql' lies in V, (q), hence r' and thus x' lies between PI' and ql', and hence x exists and lies in plq; but pir C U + pir, and rq C sq C U. A similar argument shows that for some 72 corresponding to the same 1, if p2' C V,2(p2), then (7) ,(U + p2'r') > a. This will follow when we have shown that U -4 p2r C VO(U 4+ p2'r'), or, p2r C V,(U + p2lr'). Take x C p2r, say x = g(p2, t), and put x' = g(P'2p t); then p(x, x') < 13. If x' is in p2'r', x C VP(p2'r'); if it is not, it is on C' beyond r'. Then if S2 = g(p2 t2), where t2 = /h(p2'r'), x is beyond

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S2. Now p(S2, r') < A < t, and thus s2 C V2,(q); hence, by the choice of n, s2r C s2q C Vq(q), and thus x C U, and x' C V#(U). If now we take a 6 _ 61 so small that any point in Wa(p) lies on an arc A joining W2,(q) and W2,(q*) which cuts both Vz,(p1) and V,2(p2), then on account of (6) and (7), there will be a point p' of A for which ,u(U -+ p'r') Ca. The set S of all these points p' forms a cross-section through p, as we now show. Each arc passing through Wa(p) contains just one point of S. Each point p' of S lies on an arc P1'P2' in an arc A passing through Wa(p); as the parameter distance of pi'p2' from PlP2 is < A < e/2 and PAp2 C Ve/2(P), P' C V,(p). We must show still that S is closed; this will follow if we show that for any point p' of C' in S, the points of S on all arcs passing through some Va,(p') lie in a V,'(p'). If p' is p, this follows directly from the proof we have given. If not, we replace in the proof C by C', C' by a C", Pi and pA by ps' and p4' close to p', etc. We can use the same ,B as before, when we notice one fact: there is a point q2' of C' in V0(q) such that r'q2' C U. For put q2' = g(p2', 1), where t = ,U(p2g); then q2' c Vp(q). We saw that s2 C V2,(q) and thus S2q C V,(q); but r'q2' C V,(s2q) C V2,(q). The rest of the proof follows the argument already given, and S is shown to be a cross-section. Examples.-(a) A theorem of R. L. Moore3 can be proved simply with the aid of the function ,. It is not hard to show that his family of arcs is a regular family. Map the arc AB onto the segment (0, 0), (1, 0) of the x,y-plane, and map each point p of an arc A Di of the family onto the point (x, y), where y = ,;(Aip)/Mu(A1D1), if Al was mapped onto (x, O). Evidently ju(AiDi) is a continuous function of the point Al; the rest of the argument follows from the continuity of the function g. The theorem holds if the arc AB is replaced by a more general point set. (b) We can also prove a theorem of Kerekjarto.4 A cross-section of the given closed curves is in this case seen to be an arc; a denumerable number of such arcs may be put together to form an arc from the outer curve to the innermost point. Map this arc onto the radius of a circle, and map the closed curves onto concentric circles as in the example above. * Presented to the American Mathematical Society, March 25, 1932. t NATIONAL RESEARcH FELLOW. Total beschrankt (Hansdorff). This is equivalent to asserting that

limrn n3-

n(s) = 0. co

2 p(p, q) is the distance between two points p and q, or in general, between two point sets; S(S) is the diameter of S; V,(S) consists of those points q for which p(q, S) <e, and W.(S), those points q for which p(p, S) . e. 3 R. L. Moore, Fund. Math., 4, 106-117(1923). Theorem I. ' v. Kerekjarto, Topologie I, p. 246. See also pp. 238-239. 6 I. E., any limit points of a sequence in S there may be in R lie also in S.