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6 Moore; R. L., Math. Zeitsch., 15, 254-260 (1922). I Whyburn, G. T., loc. cit., theorem 10. 8 Wilder, R. L., Fund. Math., 7, 340-77 (1925), theorem 1.

Ayres, W. L., Annals Math., Ser. 2, 28, 501-502 (1927). Whyburn, G. T., Trans. Amer. Math. Soc., 29, 369-400 (1927), theorem 7. 11 Moore, R. L., these PROCUSDINGS, 9, 101-106 (1923). 12 Moore, R. L., Bull. Amer. Math. Soc., 29, 299 (1923). 13 Knaster B., and Kuratowski, C., Fund. Math., 5, 26 (1924). 14 These PROCUSDINGS, 9, 7-12 (1923). 16 This paper is not yet published but has been submitted to the Monatsheften fur Mathematik und Physik. For an abstract, see Bull. Amer. Math. Soc., 34, 430 (1928). I

10

ON SIMPLE CLOSED CURVES AND OPEN CURVES' By W. L. Ay"S2 DUPARTMUNT OF PURE MATHEMATICS, UNIVFRS1TY or TZXAS Communicated January 5, 1929

In this note we will give a new characterization of simple closed curves and a necessary and sufficient condition in order that a continuous curve contain an open curve. The first is of interest in itself and the second has some applications in the theory of continuous curves. THUOREM 1. If no point of a continuum M in space of n dimensions is a cut point of M but every two points of M may be separated by a subset of M consisting of two points, then M is a simple closed curve. Proof.-Since every two points of M may be separated by a finite subset of M, the continuum M is a continuous curve. As M contains no cut point it is cyclicly connected,4 and contains a simple closed curve J. Let K be any component of M-J. Since M has no cut point, K has at least two limit points on J, and there exists an arc PXQ whose end points P and Q belong to J and every other point of PXQ belongs to K. Let PYQ and PZQ be the two arcs of J from P to Q. Then P and Q cannot be separated in M by two points for there are three arcs PXQ, PYQ and PZQ of M from P to Q, no two of which have any other point in common. Hence M - J is vacuous as desired. LuMMA. In order that a non-cut point P of a continuous curve M in n-dimensional space lie on a simple closed curve of M it is necessary and sufficient that M not contain a sequence of points P1, P2, P3, .. . having P as its sequential limit point and such that for every value of i, (1) M -Pi is the sum of two non-vacuous mutually separated sets, Mi and Mpi, where Mp is the set containing P, (2) Mi is a subset of Mj+,. Proof.-The condition is necessary. Suppose the non-cut point P is a point of a simple closed curve J of M. Suppose P is the sequential

VoL. 15, 1929

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limit point of a sequence P1, P2, P3, ... satisfying conditions (1) and (2). Ifi> andPi PjthenPjbelongstoM,. As J-Pisaconnected subset of M-Pi containing P, the set J- Pi belongs to Mpi for every i. Suppose some point Pk belongs to the simple closed curve J. As Pk j P and P is the sequential limit point of [Pit there is an integer m > k such that Pm d Pk. The set J-Pm contains Pk and is a subset of Mpm. But since m > k and Pm 0 Pk, Pk belongs to Mm. Hence J can contain no point Pi and J belongs to Mpi for every i. Now let co

co

H= >3Mg and K = II Mpi. i=1 i=1 If Q is any point of M which does not belong to H, then Q belongs to Pi + Mpi for every i. But for any integer k there is an integer m > k such that Pk belongs to Mm. Then Q must belong to Mpj for every i

and thus to K. Hence H + K = M. The set H is an open subset of M and thus contains no limit point of K. Every limit point of H in K is a limit point of [Pj]. Then P is the only limit point of H in K, and H and K-P are non-vacuous and mutually separated. Thus P is a cut point of M, contrary to hypothesis. The condition is sufficient. Suppose P belongs to no simple closed curve of M. Let Q be any other point of M and PQ be an arc of M from P to Q. The point P is a limit point of points of PQ which separate P and Q in M.5 Let P1, P2, P3, . . . be a sequence of such points of PQ having P as a sequential limit point and such that the subarc PP, of PQ contains Pi+1. For every i let Mi denote the component of M-Pi containing the subset P5Q-Pi of PQ, and let Mpi be M-Pi-Mi. The sets Mi and Mpi are mutually separated and for every value of i, Mi is a subset of Mi+,. But by hypothesis M contains no such sequence of points [Pi]. Thus the condition is sufficient. THZORZM 2. In order that an unbounded continuous curve M in n dimensions contain an open curve it is necessary and sufficient that M not contain a sequence of points P1, P2, P3, . . . such that (1) the sequence [Pji has no limit point, (2) for every i, M-Pi is the sum of two non-vacuous mutually separated sets Mu and Mb, (3) for every i, Mb is bounded and is a subset of Mib+l. Proof.-The condition is necessary. Suppose M contains an open curve C. If M is the entire space, the condition is satisfied obviously. If not, let T be an inversion whose center P does not belong to M. Then P is a non-cut point of the continuous curve T(M) + P lying on the simple closed curve T(C) + P. Now suppose M does contain a sequence [Pi] satisfying (1), (2) and (3) of the theorem. Then [T(Pi)] is a sequence having P as a sequential limit point. From condition (2), T(Mib) and T(M') + P are mutually separated. And from condition (3) we

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have that T(Mb) is a subset of T(Mbi6). But by the Lemma, T(M) + P can contain no such sequence [T(Pi)]. The condition is sufficient. If M is the entire space, M contains an open curve. If not, let T be an inversion with center P not belonging to M. If T(M) + P contains a sequence [Pi] satisfying conditions (1) and (2) of the Lemma, then the sequence [T-'(Pi)] satisfies the three conditions of the statement of our theorem. But by hypothesis there is no such sequence [T-'(Pi)]. Then the hypothesis of the Lemma is satisfied and P lies on some simple closed curve J of T(M) + P. The set T-1(J-P) is an open curve of M. 1 Presented to the American Mathematical Society, November 26, 1927, as a portion of the preceding paper. 2 NATIONAL REsEARCH FULLOW IN MAmTMATIcs. 3G. T. Whyburn and W. L. Ayres, Bull. Amer. Math. Soc., 34, 349-360 (1928), theorem 2. 4 G. T. Whyburn, these PROCSSDINGS, 13, 31-38, 1927. All the theorems of this paper, except theorem 10, are extended to n dimensions in my paper, "Concerning Continuous Curves in Space of n Dimensions," which will appear in the American Journal of Mathematics. 6 W. L. Ayres, these PROCEEDINGS, 13, 749-754, 1927, theorem 1. This theorem is extended to n dimensions in my paper mentioned in reference 4.

ON A RELATION BETWEEN CONFORMAL AND PROJECTIVE GROUPS IN FUNCTION SPACE By I. A. BmRNET DUPARTM13NT OF MATHOMATICS, THI UNIVURSITY OF CINCdNNArI Communicated December 26, 1928

In an important memoir' published in 1871, Felix Klein pointed out an interesting relation between the general conformal group in n-space and the projective group of the sphere in n + 1 space. His result may be stated as follows: The continuous group of projective transformations in n + 1 space, which leave invariant the unit sphere in this space, may by a stereographic projection be transformed into the general conformal group in n-space. It is the purpose of this note to show that there is an analogous relation between the projective and conformal transformations in certain types of function spaces which will be defined presently. Not only is it interesting to note the extension of Klein's result, but the method used in the present paper leads to the consideration of a new class of infinitesimal transformations which are a generalization of those used by Kowalewski.2 Let Rx denote the space of continuous functions, so that a point of