An indecomposable limit sum

N. E. Rutt
1937 Bulletin of the American Mathematical Society  
It is the object of this paper to investigate a certain simple monotone sequence of continua. The theorem of the paper states conditions under which the limit sum of the sequence is indecomposable. The precise formulation and proof of the theorem will be undertaken after the following lemma is established. LEMMA. Let K be a plane bounded indecomposable continuum and L a plane bounded continuum such that K-L^O, and that c(L)* includes a particular component X containing the component b of c(L+K)
more » ... with the following properties: (a) the set L contains two distinct points, a and c, connected through b by the arc B which divides b into bi and b e , and X into X; and X e ; (b) both X» and X e contain points of K. Then each component of c(K+L) has as its boundary a proper subset of K+L. The assumption that c(K+L) has a component y with boundary T such that T 3 (K+L) will be shown contradictory. Let the boundaries of S*, ô e , X;, X e be respectively A t -, A e , A»-, and A e . Suppose that b is unbounded and also b e and X e , so that 8i and X; will necessarily be bounded. Evidently X; 3 8i and X e 3 b e . Consider first the case in which L is irreducible between a and c. Both Ai and A e contain L. For Ai^L+B and A e cL+B; so, since B is an arc with L-(.B)f = 0, AiL and A e L are continua containing a+c. If either of these is not identical with L, then L is reducible between a and c. The domains b and y are, moreover, identical, for both X» and X e contain points of K, therefore points of T, and therefore points of 7. There is thus an arc X in y such that X-X^O and XX e^0 , and since XL = 0, then (B)-X5*0. This implies X-ô^O, accordingly 7-6^0; and as both 7 and b are components of c(K+L), then 7 = 5, and \+LDTDK. Let Ki be the sum of K\i and of all the components of L-K * If X is a point set then c(X) is the complement of X. f If X is an arc then (X) is X with ends omitted.
doi:10.1090/s0002-9904-1937-06626-2 fatcat:z7xpcp6w3vfhvlepienppz3hry