. Transactions of the American Mathematical Society . ll disconnect S is that M, the set of points common to M\. andJtf 2, be not connected. Proof. The condition is necessary. For suppose that M is connected;we can prove that M does not disconnect S . For suppose S — J/ = Si + <S2where 52 denotes the point set P0 B2 - P0 - £2. 1922] PLANE CONNECTED POINT SETS 155 with PI in a connected subset of Si. It will be seen that since M is closed,these points form a domain Ds, a subset of Si, and of the interior of S, suchthat the_ boundary of Ds is a subset of M_. Let M denote the point setM! -- 1/
. Transactions of the American Mathematical Society . ll disconnect S is that M, the set of points common to M\. andJtf 2, be not connected. Proof. The condition is necessary. For suppose that M is connected;we can prove that M does not disconnect S . For suppose S — J/ = Si + <S2where 52 denotes the point set P0 B2 - P0 - £2. 1922] PLANE CONNECTED POINT SETS 155 with PI in a connected subset of Si. It will be seen that since M is closed,these points form a domain Ds, a subset of Si, and of the interior of S, suchthat the_ boundary of Ds is a subset of M_. Let M denote the point setM! -- 1/2 and M the point set J/2 - M. Let P2 denote a point of S2that is without S; by the lemma, P2 can be joined to PI by an arc not containing any point of M2. Let P denote the first point the arc P2 PI has on. the boundary of Ds (Fig. 5), let P2 be joined to PI by an arc not containingany point of MI , and let P denote the first point of this arc P2 PI on theboundary of D,. Then P belongs to M and P to M. Then there existsan arc PP3 P, a subset of P2 P + P2 P, which lies, except for its endpoints, without Ds. As in Theorem 3, there exists a simple closed curve Hienclosing P, containing only one point LI of the arc PP3 P, and neithercontaining nor enclosing a point of J/2, and there exists a correspondingcurve HI enclosing P, containing only one point L2 of the arc PP3 P andneither containing nor enclosing a point of MI or HI , and there exists an arc C\ R\in Ds having only C\ on HI and RI on Hz. Then there is a simple closed curve«/i, composed of LI P3 L2 +_ Ci RI together with the arcs LI C\ on HI andL2 RI on H2 so chosen that Ji encloses P and P. We can show as in Theorem 3 that if those points of the boundary of Dsthat this curve contains or encloses belong to M + M then two points,one
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