Concerning the Division of the Plane by Continua

176

MA THEMA TICS: J. R. KLINE

PROC. N. A. S.

CONCERNING THE DIVI-SION OF THE PLANE BY CONTINUA By JOHN ROBERT KLINE DEPARTMZNT OF MATHUMATICS, UNIVURSITY OF PI3NNSYLVANIA Communicated April 1, 1924

In a paper recently published in the Fundamenta Mathenaticae,' I proved that if M is a closed connected set which contains more than one point and remains connected (in the weak sense),2 upon the removal of any connected subset g, then M is a simple closed curve. In the same number of the Fundamenta, Kuratowski3 proved that if C is a bounded continu, which remains connected in the strong sense upon the removal of any subcontinu, then C is a simple closed curve. Kuratowski's theorem would not be true provided it were merely required that C remain connected in the weak sense upon the removal of any subcontinu. For consider the set M = 4 Mi, where M1 is the point set composed of y = sin 1/x for o<x<1/ir, M2 the interval (1/7r, o) to (1/r, -2), Ms is the interval (1/r, -2) to (0, -2) while M4 is the interval (0, -2) to (0, +1). The set M does not remain connected in the strong sense provided the interval from (0, 0) to (0, -2) is removed. But the set M remains connected in the weak sense no matter what subcontinu is removed. In the present paper I shall consider bounded decomposable4 continua lying in a plane S, and having the property that they remain connected in the weak sense upon the removal of any sub-continu. I shall prove that any continu satisfying these conditions divides its plane and that unless it is a simple closed curve, it cannot be a continuous curve., THEOREM A. Suppose M is a bounded decomposable continu lying in a plane S such that if g is any subcontinu of M, then M-g is connected.6 Under these hypotheses, M disconnects its plane S. Proof: Suppose M does not disconnect its plane. Now as M is decomposable, then M = M1 + M2, two continua, each different from M. Hence neither of the sets, M1 and M2, is a proper subset of the other one. Neither M1 nor M2 can divide the plane S. For, suppose S -M = GI + G2, two mutually exclusive sets neither of which contains a limit point of the other one. Now, by supposition, S - (M1 + M2), a subset of S M1, is connected. Hence S - (M1 + M2) must be a subset either of G1 or of G2: Let us suppose it is a subset of G1. Then G2 is a subset of M2. Let P be any point of G2. As no point of G2 is a limit point of G1, there is a circle C enclosing P such that neither C nor its interior contains a point of GC. Hence neither C nor its interior contains a point of S - (M1 + M2). Hence C plus its interior is a subset of M. Let J be any simple closed curve lying entirely within C and hence belonging to M. As there are points of M within J and also points of M without J, then J is a continu whose removal fails to leave M connected in the weak sense. Hence the supposition that M1 disconnects the plane has lead to a contradiction.

VOL. 10, 1924

MA THEMA TICS: J. R. KLINE

177

In like manner M2 fails to disconnect S. Several cases may arise: Case i. D, the common part of M1 and M2 is connected. Then as M1 and M2 are closed, D is a continu. Then M - D = (M1 - D) + (M2- D). Neither of these sets is vacuous nor can either contain a limit point of the other one. Hence D divides M, contrary to hypothesis. Case II. D, the common part of M1 and M2 is not connected. Then as neither M1 nor M2 divides S, it follows from a theorem due to Miss Anna Mullikin,7 that M1 + M2 must divide S into at least two parts. THEOREM B. Suppose that M is a bounded decomposable plane continu such that if g is any subcontinu of M, then M-g is connected in the weak sense. Under these hypotheses, if M is a continuous curve, then M is a simple closed curve. Proof. Suppose M is a bounded continu satisfying the hypotheses of our theorem. Let g be any subcontinu of M. Then M-g satisfied R. L. Moore's definition of a maximal connected subset of the open subset M-g.8 Hence by a theorem due to Moore, M-g is arcwise connected. As an arc is a continu, then any two points of M-g lie together in a closed and connected subset of M-g. Hence M-g is connected in the strong sense. Hence by the theorem due to Kuratowski, M is a simple closed curve. 1 Cf. my paper "Closed Connected Sets which Remain Connected upon the Removal of Certain Connected Subjects." Fund. Math., 5, pp. 3-10, 1924. 2 A point set M is said to be connected (in the weak sense) if however, it be divided into two mutually exclusive sets, one of them contains a limit point of the other one. A continu is a set that is both closed and connected. A set M is said to be connected in the strong sense (semi continu in terminology of Kuaster and Kuratowski) if every pair of points A and B belonging to M, lie together in a closed connected subset of M. 3 Cf. C. Kuratowski, "Contribution a L'etude de continus de Jordan," Fund. Math., 5, p. 119, 1924. 4A, continu which is not the sum of two continua each different from itself is said to be indecomposable. Every indecomposable continu M is such that if g is a subcontinu of M, then M-g is connected. It is evident that it is possible to construct an indecomposable plane continu which is bounded and fails to divide its plane. 6 A continuous curve is a closed connected set M which is connected im kleinen at every one of its points. 6 Here after in this paper, unless otherwise stated, "connected" and "connected in the weak sense" will be considered as synomymous terms. 7 Cf. Anna M. Mullikin, "Certain Theorems Relating to Plane Connected Sets," Trans. Amer. Math. Soc., 24, p. 154, 1922. Miss Mullikin's result is as follows: If M, and M, are two closed connected bounded point sets neither of which disconnects a plane S, then a necessary and sufficient condition that their sum M shall disconnect S is that M, the set of points common to M1 and M2 be not connected. 8 Cf. R. L. Moore, "Report on Continuous Curves from the Viewpoint of Analysis Situs," Bull. Amer. Math. Soc., 29, p. 294, 1923. According to Moore, a point set M is said to be arcwise connected if every two distinct points of M are the extremities of a simple continuous arc which lies wholly in M. The point set K is said to be an open subset of M if K is a subset of M and M-K is either vacuous or closed. A maximal connected subset of a point set K is a connected subset of K which is not a proper subset of any other connected subset of K. Moore's result is as follows: In order that a continuum M should be a continuous curve it is necessary that every maximal connected subset of an open subset of M should be arcwise connected.