A CONTINUUM-CHAINABLE APOSYNDETIC PLANE CONTINUUM

ABSTRACT. A continuum is said to be conthzuum-chainable provided, fbr each pair x,y of points and each e> 0, there exists a chain of subcontinua from x to y such that each link has diameter less than e. It is known that continuum-chainability and arcwise connectivity are equivalent in each of the following classes of continua: non-separating plane continua, continua irreducible about a finite set, atriodic continua and hereditarily unicoherent continua. In this paper, we construct a continuum-chainable aposyndetic plane continuum which is not arcwise connected. This answers a question by C. L. Hagopian and L. E. Rogers. I. Definitions and preliminary remarks. By a continuum we mean a compact connected metric space. A finite collection {C 1,C2,...,C n} of sets is a chain from x to y provided x belongs to C 1, y belongs to C n and, for i,j G ( 1....,n}, C i N Cj 4:0 if and only if li-jl •< 1. Let e 2> 0. By an e-continut•m-chain we mean a chain such that each link Cj is a subcontinuum of M with diameter less than e. Thus a continuum M is continuum chainable if and only if, for each pair of points x,y G M and each e 2> 0, there exists an e-continuum-chain from x to y. Continumn-chainability is a generalization of property S (see [4], page 20) which is equivalent to local connectedness (ibidem, page 23) for continua. It was introduced by Ira Rosenholtz. A continuum M is aposyndetic at x with respect to y (see [3]) provided there exists a subcontinuum H of M such that x C H ¸ (the interior of H) and y • H. If M is aposyndetic at each x G M with respect to each y C M\{x}, then we say that M is aposyndetic. A continuuln M is said to be semi-aposyndetic if for each pair of distinct points x and y of M, M is aposyndetic either at x with respect to y or at y with respect to x. In [1] (the author is indebted to C. L. Hagopian for tiffs reference), Hagopian showed that each semi-aposyndetic E-continuum (i.e., a plane continuum such that for each e> 0, there are at most a finite nmnber of complementary domains of M o