Aposyndetic properties of unicoherent continua

In the first part of this paper the structure of w-aposyndetic continua is studied. In particular, those continua which are w-aposyndetic but fail to be in + l)-aposyndetic are investigated. Unicoherence is shown to be a sufficient condition for an %-aposyndetic continuum to be in + l)-aposyndetic. In the final portion of the paper a stronger form of unicoherence is defined. As a point-wise property, aposyndesis and connected im kleinen are shown to be equivalent in continua with this property. Throughout this paper a continuum is a compact connected metric space and M will denote a continuum. If N is a subcontinuum of M, the interior of N in M will be denoted by int N. Suppose pe M and F is a closed subset of M such that p ί F. M is aposyndetic at p with respect to F if there is a subcontinuum N of M such that p e int N c N c M — F. Let n be a positive integer. If M is aposyndetic at p with respect to each subset of M consisting of n points, then M is n-aposyndetic at p. M is n-aposyndetic if it is ^-aposyndetic at each point. By convention if M is 1-aposyndetic then M is said to be aposyndetic. For other terms not defined herein, see [3], [4] and [6]. LEMMA 1. Suppose M is n-aposyndetic, peM,F is a subset of M — {p} consisting of n + 1 points, and M is not aposyndetic at p with respect to F. If F 1 and F2 are disjoint nonempty subsets of F such that F = Fι U F21 there exist subcontinua H and K such that Fιa H — K, F2 c K- H,pemt(HΓ\K), and M= H\JK.